35:
68:
2417:
nilpotent Lie group (equivalently it does not contain a nontrivial compact subgroup) then a discrete subgroup is a lattice if and only if it is not contained in a proper connected subgroup (this generalises the fact that a discrete subgroup in a vector space is a lattice if and only if it spans the
4481:
The real rank of a Lie group has a significant influence on the behaviour of the lattices it contains. In particular the behaviour of lattices in the first two families of groups (and to a lesser extent that of lattices in the latter two) differs much from that of irreducible lattices in groups of
1062:
is a matter of what it is designed to achieve. Maybe the most obvious idea is to say that a subgroup "approximates" a larger group is that the larger group can be covered by the translates of a "small" subset by all elements in the subgroups. In a locally compact topological group there are two
6701:(the group of 2 by 2 matrices with determinant 1) any isomorphism of lattices is essentially induced by an isomorphism between the groups themselves. In particular, a lattice in a Lie group "remembers" the ambient Lie group through its group structure. The first statement is sometimes called
6267:
2353:
Not every locally compact group contains a lattice, and there is no general group-theoretical sufficient condition for this. On the other hand, there are plenty of more specific settings where such criteria exist. For example, the existence or non-existence of lattices in
4737:
The property known as (T) was introduced by
Kazhdan to study the algebraic structure lattices in certain Lie groups when the classical, more geometric methods failed or at least were not as efficient. The fundamental result when studying lattices is the following:
2837:
The criterion for nilpotent Lie groups to have a lattice given above does not apply to more general solvable Lie groups. It remains true that any lattice in a solvable Lie group is uniform and that lattices in solvable groups are finitely presented.
7050:
It is easily seen from the basic theory of group actions on trees that uniform tree lattices are virtually free groups. Thus the more interesting tree lattices are the non-uniform ones, equivalently those for which the quotient graph
3172:
2336:
1145:
3811:
6142:
6598:
2380:. This is sufficient to imply the existence of a lattice in a Lie group, but in the more general setting of locally compact groups there exist simple groups without lattices, for example the "Neretin groups".
6541:
5115:
3361:
4752:
it is possible to classify semisimple Lie groups according to whether or not they have the property. As a consequence we get the following result, further illustrating the dichotomy of the previous section:
1075:, so it gives finite mass to compact subsets, the second definition is more general. The definition of a lattice used in mathematics relies upon the second meaning (in particular to include such examples as
6721:
provides (for Lie groups and algebraic groups over local fields of higher rank) a strengthening of both local and strong rigidity, dealing with arbitrary homomorphisms from a lattice in an algebraic group
3052:
1903:
1060:
3679:
3596:
4840:. For uniform lattices this is a direct consequence of cocompactness. In the non-uniform case this can be proved using reduction theory. It is easier to prove finite presentability for groups with
3407:
4821:
4726:
4590:
3466:
6835:
6782:
6699:
3739:
2929:
2264:
2221:
6440:
5811:
A class of groups with similar properties (with respect to lattices) to real semisimple Lie groups are semisimple algebraic groups over local fields of characteristic 0, for example the
1453:
4642:
3858:
1316:
6022:
5205:
3972:
3925:
7157:
2829:
Finally, a nilpotent group is isomorphic to a lattice in a nilpotent Lie group if and only if it contains a subgroup of finite index which is torsion-free and finitely generated.
7075:
5757:
5397:
3275:
2737:
2393:
For nilpotent groups the theory simplifies much from the general case, and stays similar to the case of
Abelian groups. All lattices in a nilpotent Lie group are uniform, and if
4524:
4345:
4240:
4132:
1533:
7364:
7045:
7001:
6940:
5257:
6081:
5919:
5890:
5841:
5711:
3512:
2077:
2048:
1829:
1767:
1580:
1001:
972:
6471:
5553:
4929:
2761:
2599:
2555:
2511:
2463:
7476:
7430:
1018:
Rigorously defining the meaning of "approximation of a continuous group by a discrete subgroup" in the previous paragraph in order to get a notion generalising the example
6397:
6352:
5283:
5025:
4026:
3201:
2997:
2951:
513:
488:
451:
7281:, all of whose vertex groups are finite, and under additional necessary assumptions on the index of the edge groups and the size of the vertex groups, then the action of
6134:
5142:
4448:
1855:
6107:
2172:
1971:
1799:
1671:
1407:
5843:. There is an arithmetic construction similar to the real case, and the dichotomy between higher rank and rank one also holds in this case, in a more marked form. Let
7299:
7275:
5504:
5323:
5166:
4407:
4302:
4194:
3878:
3619:
3536:
3080:
2824:
2675:
2635:
2144:
1991:
1923:
1691:
1623:
1336:
1204:
1499:
6302:
6052:
5674:
4905:
4475:
4381:
4276:
4168:
4077:
1251:
1379:
4999:
815:
7496:
7450:
7404:
7384:
7327:
7237:
7217:
7197:
7177:
7103:
6960:
6895:
6875:
6491:
6375:
6322:
5951:
5861:
5797:
5777:
5731:
5637:
5617:
5597:
5577:
5484:
5464:
5437:
5417:
5371:
5343:
5303:
5225:
4973:
4949:
4875:
4050:
4004:
3486:
3295:
3229:
3100:
2975:
2882:
2784:
2695:
2655:
2575:
2531:
2483:
2439:
2411:
2124:
2104:
2011:
1943:
1738:
1718:
1643:
1603:
1473:
1356:
1271:
1224:
1184:
6966:, in which a basis of neighbourhoods of the identity is given by the stabilisers of finite subtrees, which are compact). Any group which is a lattice in some
7243:
which in this case is a tree. In contrast to the characteristic 0 case such lattices can be nonuniform, and in this case they are never finitely generated.
3105:
2269:
1078:
3744:
860:
as a periodic subset of points, and both the algebraic structure of lattices and the geometry of the space of all lattices are relatively well understood.
945:
Lattices are best thought of as discrete approximations of continuous groups (such as Lie groups). For example, it is intuitively clear that the subgroup
6262:{\displaystyle \mathrm {SL} _{2}\left(\mathbb {Z} \left\right)\subset \mathrm {SL} _{2}(\mathbb {R} )\times \mathrm {SL} _{2}(\mathbb {Q} _{p})}
7106:
373:
5759:
are called locally symmetric spaces. There is thus a bijective correspondence between complete locally symmetric spaces locally isomorphic to
8127:
8137:
Gelander, Tsachik (2014). "Lectures on lattices and locally symmetric spaces". In
Bestvina, Mladen; Sageev, Michah; Vogtmann, Karen (eds.).
6837:
are not locally rigid. In fact they are accumulation points (in the
Chabauty topology) of lattices of smaller covolume, as demonstrated by
6553:
323:
7498:), more general than the stronger condition that the quotient be finite (as proven by the very existence of nonuniform tree lattices).
6631:
results state that in most situations every subgroup which is sufficiently "close" to a lattice (in the intuitive sense, formalised by
6496:
5030:
3300:
808:
318:
8077:
8047:
8024:
7984:
7810:
2373:. It is also not very hard to find unimodular groups without lattices, for example certain nilpotent Lie groups as explained below.
6442:
of adélic points is well-defined (modulo some technicalities) and it is a locally compact group which naturally contains the group
3002:
6639:) is actually conjugated to the original lattice by an element of the ambient Lie group. A consequence of local rigidity and the
1860:
8188:
4837:
2342:
1021:
8178:
3624:
3541:
1158:. Uniform lattices are quasi-isometric to their ambient groups, but non-uniform ones are not even coarsely equivalent to it.
734:
3366:
4760:
4665:
4529:
5350:
4655:
while all normal subgroups of irreducible lattices in higher rank are either of finite index or contained in their center;
801:
6605:
6547:
1012:
3419:
6794:
6741:
232:
6661:
3688:
2891:
2226:
2183:
6600:
to more classical S-arithmetic quotients. This fact makes the adèle groups very effective as tools in the theory of
6277:. The Margulis arithmeticity theorem applies to this setting as well. In particular, if at least two of the factors
2841:
Not all finitely generated solvable groups are lattices in a Lie group. An algebraic criterion is that the group be
864:
6406:
8173:
7514:
Bader, Uri; Caprace, Pierre-Emmanuel; Gelander, Tsachik; Mozes, Shahar (2012). "Simple groups without lattices".
4084:
1412:
1068:
845:
7240:
4595:
6640:
2791:
2787:
1004:
616:
350:
227:
115:
42:, a discrete subgroup of the continuous Heisenberg Lie group. (The coloring and edges are only for visual aid.)
28:
6730:. It was proven by Grigori Margulis and is an essential ingredient in the proof of his arithmeticity theorem.
3816:
1276:
8183:
7478:
be of "finite volume" in a suitable sense (which can be expressed combinatorially in terms of the action of
6838:
6738:
The only semisimple Lie groups for which Mostow rigidity does not hold are all groups locally isomorphic to
6655:
6493:-rational point as a discrete subgroup. The Borel–Harish-Chandra theorem extends to this setting, and
5978:
5171:
4833:
3930:
3883:
2366:
7118:
5443:
2885:
898:
766:
556:
7252:
7054:
5736:
5376:
3234:
2700:
883:
6963:
4743:
A lattice in a locally compact group has property (T) if and only if the group itself has property (T).
4489:
4413:
4310:
4205:
4097:
2365:. This allows for the easy construction of groups without lattices, for example the group of invertible
1504:
906:
837:
640:
7332:
7013:
6969:
6908:
6621:
Another group of phenomena concerning lattices in semisimple algebraic groups is collectively known as
5233:
2341:
Any finite-index subgroup of a lattice is also a lattice in the same group. More generally, a subgroup
6057:
5895:
5866:
5817:
5690:
3491:
2053:
2024:
1808:
1746:
1538:
977:
948:
8152:
6785:
6445:
5534:
4910:
4053:
2742:
2580:
2536:
2492:
2444:
580:
568:
186:
120:
8193:
7455:
7409:
5677:
4659:
2486:
887:
882:
Lattices are also well-studied in some other classes of groups, in particular groups associated to
871:. In particular there is a wealth of rigidity results in this setting, and a celebrated theorem of
155:
50:
7846:
Gelander, Tsachik (15 September 2004). "Homotopy type and volume of locally symmetric manifolds".
6380:
6335:
5262:
5004:
4009:
3203:
a consequence of the arithmetic construction is that any semisimple Lie group contains a lattice.
3184:
2980:
2934:
2361:
As we mentioned, a necessary condition for a group to contain a lattice is that the group must be
1071:) or measure-theoretical (a subset of finite Haar measure). Note that since the Haar measure is a
496:
471:
434:
8142:
7957:
7904:
7855:
7783:
7757:
7659:
7549:
7523:
7110:
6112:
5120:
4418:
918:
140:
112:
8070:
Algebraic groups and number theory. (Translated from the 1991 Russian original by Rachel Rowen.)
7432:
being finite. The general existence theorem is more subtle: it is necessary and sufficient that
4823:
do not have
Kazhdan's property (T) while irreducible lattices in all other simple Lie groups do;
1834:
6086:
2149:
1948:
1776:
1648:
1384:
8123:
8073:
8065:
8043:
8020:
7980:
7806:
7010:
The discreteness in this case is easy to see from the group action on the tree: a subgroup of
6636:
6632:
5640:
4952:
4749:
4243:
849:
841:
711:
545:
388:
282:
7284:
7260:
5489:
5308:
5151:
4386:
4281:
4173:
3863:
3604:
3521:
3065:
2797:
2660:
2608:
2129:
1976:
1908:
1676:
1608:
1321:
1189:
8015:
Tree lattices With appendices by H. Bass, L. Carbone, A. Lubotzky, G. Rosenberg, and J. Tits
7941:
7914:
7865:
7767:
7643:
7533:
6601:
5681:
4092:
2859:
2842:
2414:
2362:
2080:
1478:
910:
876:
872:
696:
688:
680:
672:
664:
652:
592:
532:
522:
364:
306:
181:
150:
39:
8108:
8087:
8057:
7994:
7953:
7820:
7779:
7686:
7655:
7545:
6280:
6030:
5645:
4883:
4453:
4354:
4249:
4141:
4059:
1229:
863:
The theory is particularly rich for lattices in semisimple Lie groups or more generally in
8104:
8100:
8083:
8053:
8039:
7990:
7949:
7816:
7775:
7682:
7651:
7628:
7541:
7278:
6902:
5526:
4652:
4029:
3983:
2954:
2602:
1364:
1151:
857:
780:
773:
759:
716:
604:
527:
357:
271:
211:
91:
24:
6844:
As lattices in rank-one p-adic groups are virtually free groups they are very non-rigid.
4978:
8156:
7301:
on the Bass-Serre tree associated to the graph of groups realises it as a tree lattice.
7481:
7435:
7389:
7369:
7312:
7222:
7202:
7182:
7162:
7088:
6945:
6880:
6860:
6628:
6476:
6400:
6360:
6307:
5936:
5846:
5782:
5762:
5716:
5622:
5602:
5582:
5562:
5469:
5449:
5422:
5402:
5356:
5328:
5288:
5210:
4958:
4934:
4860:
4135:
4035:
3989:
3471:
3280:
3214:
3085:
3059:
2960:
2867:
2769:
2680:
2640:
2560:
2516:
2468:
2424:
2396:
2109:
2089:
1996:
1928:
1723:
1703:
1628:
1588:
1458:
1341:
1256:
1209:
1169:
1155:
926:
914:
909:(through the construction of locally homogeneous manifolds), in number theory (through
902:
787:
723:
413:
393:
330:
295:
216:
206:
191:
176:
130:
107:
6713:(Mostow proved it for cocompact lattices and Prasad extended it to the general case).
3167:{\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )\subset \mathrm {SL} _{2}(\mathbb {R} )}
2331:{\displaystyle \mathrm {SL} _{n}(\mathbb {Z} )\subset \mathrm {SL} _{n}(\mathbb {R} )}
1140:{\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )\subset \mathrm {SL} _{2}(\mathbb {R} )}
34:
8167:
7961:
7553:
6898:
6717:
6706:
5510:
4878:
4200:
3806:{\displaystyle \mathrm {SL} _{2}(\mathbb {R} )\times \mathrm {SL} _{2}(\mathbb {R} )}
2178:
1359:
1072:
922:
706:
628:
462:
335:
201:
7787:
7663:
5960:
then there are uncountably many commensurability classes of non-arithmetic lattices.
5486:
and we see that the quotient manifold is of finite
Riemannian volume if and only if
1696:
In the case of discrete subgroups this invariant measure coincides locally with the
8013:
6710:
6647:
there are only finitely many (up to conjugation) lattices with covolume bounded by
6355:
5812:
5556:
4841:
4645:
3055:
2377:
2370:
1720:
being a lattice is equivalent to it having a fundamental domain (for the action on
1697:
929:
561:
260:
249:
196:
171:
166:
125:
96:
59:
7869:
6784:. In this case there are in fact continuously many lattices and they give rise to
3514:
is said to be irreducible if either of the following equivalent conditions hold:
1585:
A slightly more sophisticated formulation is as follows: suppose in addition that
7932:
Lubotzky, Alexander (1991). "Lattices in rank one Lie groups over local fields".
4844:; however, there is a geometric proof which works for all semisimple Lie groups.
8072:. Pure and Applied Mathematics. Vol. 139. Boston, MA: Academic Press, Inc.
7113:
5514:
5346:
5145:
1625:
is discrete it is also unimodular and by general theorems there exists a unique
1064:
868:
5579:
is not compact it is not definite and hence not an inner product: however when
4079:). The semisimple Lie groups of real rank 1 without compact factors are (up to
3181:
in a semisimple Lie group. Since all semisimple Lie groups can be defined over
1147:) but the first also has its own interest (such lattices are called uniform).
5964:
In the latter case all lattices are in fact free groups (up to finite index).
4348:
829:
728:
456:
7918:
6608:
are usually stated and proven for adélic groups rather than for Lie groups.
2355:
1008:
549:
7977:
Adeles and algebraic groups. With appendices by M. Demazure and
Takashi Ono
6273:
This arithmetic construction can be generalised to obtain the notion of an
7537:
67:
6643:
is Wang's theorem: in a given group (with a fixed Haar measure), for any
5800:
4351:
coefficients which preserve a "quaternionic quadratic form" of signature
2177:
All of these examples are uniform. A non-uniform example is given by the
86:
7805:. Progress in Mathematics. Vol. 212. Birkhäuser Verlag. Chapter 7.
6658:
states that for lattices in simple Lie groups not locally isomorphic to
5509:
Interesting examples in this class of
Riemannian spaces include compact
7979:. Progress in Mathematics. Vol. 23. Birkhäuser. pp. iii+126.
7945:
7771:
7647:
7386:
contain a lattice? The existence of a uniform lattice is equivalent to
4080:
428:
342:
4450:(the real form of rank 1 corresponding to the exceptional Lie algebra
7895:
Gelander, Tsachik (December 2011). "Volume versus rank of lattices".
7860:
7762:
7047:
is discrete if and only if all vertex stabilisers are finite groups.
7105:
is a local field of positive characteristic (i.e. a completion of a
6593:{\displaystyle \mathrm {G} (F)\backslash \mathrm {G} (\mathbb {A} )}
1003:
in some sense, while both groups are essentially different: one is
8147:
7909:
7528:
4658:
Conjecturally, arithmetic lattices in higher-rank groups have the
2794:
since it is itself nilpotent); in fact it is generated by at most
1831:
is a uniform lattice if and only if there exists a compact subset
1206:
a discrete subgroup (this means that there exists a neighbourhood
6625:. Here are three classical examples of results in this category.
6536:{\displaystyle \mathrm {G} (F)\subset \mathrm {G} (\mathbb {A} )}
5110:{\displaystyle g_{\gamma }(v,w)=g_{e}(\gamma ^{*}v,\gamma ^{*}w)}
3356:{\displaystyle \Gamma _{1}\subset G_{1},\Gamma _{2}\subset G_{2}}
7109:
of a curve over a finite field, for example the field of formal
7077:
is infinite. The existence of such lattices is not easy to see.
6877:
be a tree with a cocompact group of automorphisms; for example,
5799:. This correspondence can be extended to all lattices by adding
2485:
can be defined over the rationals. That is, if and only if the
2079:. A slightly more complicated example is given by the discrete
7748:
Raghunathan, M. S. (2004). "The congruence subgroup problem".
6136:
is a real Lie group. An example of such a lattice is given by
5972:
More generally one can look at lattices in groups of the form
5779:
and of finite
Riemannian volume, and torsion-free lattices in
5733:
if and only if it is discrete and torsion-free. The quotients
3685:
An example of an irreducible lattice is given by the subgroup
3177:
Generalising the construction above one gets the notion of an
3047:{\displaystyle \Gamma =G\cap \mathrm {GL} _{n}(\mathbb {Z} )}
1740:
by left-translations) of finite volume for the Haar measure.
8117:
7462:
7416:
7061:
6571:
5743:
5383:
4651:
Lattices in rank 1 Lie groups have infinite, infinite index
4648:) but all irreducible lattices in the others are arithmetic;
3409:
is a lattice as well. Roughly, a lattice is then said to be
2533:
is a nilpotent simply connected Lie group whose Lie algebra
1898:{\displaystyle G=\bigcup {}_{\gamma \in \Gamma }\,C\gamma }
5619:
is a maximal compact subgroup it can be used to define a
4848:
Riemannian manifolds associated to lattices in Lie groups
4056:
elements with at least one real eigenvalue distinct from
1700:
and hence a discrete subgroup in a locally compact group
1063:
immediately available notions of "small": topological (a
1055:{\displaystyle \mathbb {Z} ^{n}\subset \mathbb {R} ^{n}}
1011:, while the other is not finitely generated and has the
16:
Discrete subgroup in a locally compact topological group
3674:{\displaystyle G_{i_{1}}\times \ldots \times G_{i_{k}}}
3591:{\displaystyle G_{i_{1}}\times \ldots \times G_{i_{k}}}
2021:
The fundamental, and simplest, example is the subgroup
897:
Lattices are of interest in many areas of mathematics:
875:
states that in most cases all lattices are obtained as
7309:
More generally one can ask the following question: if
3402:{\displaystyle \Gamma _{1}\times \Gamma _{2}\subset G}
974:
of integer vectors "looks like" the real vector space
7484:
7458:
7438:
7412:
7392:
7372:
7335:
7315:
7287:
7263:
7225:
7205:
7185:
7165:
7121:
7091:
7057:
7016:
6972:
6948:
6911:
6883:
6863:
6797:
6744:
6664:
6556:
6499:
6479:
6448:
6409:
6383:
6363:
6338:
6310:
6283:
6145:
6115:
6089:
6060:
6033:
5981:
5939:
5898:
5869:
5849:
5820:
5785:
5765:
5739:
5719:
5693:
5648:
5625:
5605:
5585:
5565:
5537:
5492:
5472:
5452:
5425:
5405:
5379:
5359:
5331:
5311:
5291:
5265:
5236:
5213:
5174:
5154:
5123:
5033:
5007:
4981:
4961:
4937:
4913:
4886:
4863:
4816:{\displaystyle \mathrm {SO} (n,1),\mathrm {SU} (n,1)}
4763:
4721:{\displaystyle \mathrm {SO} (n,1),\mathrm {SU} (n,1)}
4668:
4598:
4585:{\displaystyle \mathrm {SU} (2,1),\mathrm {SU} (3,1)}
4532:
4492:
4456:
4421:
4389:
4357:
4313:
4284:
4252:
4208:
4176:
4144:
4100:
4062:
4038:
4012:
3992:
3933:
3886:
3880:
is the Galois map sending a matric with coefficients
3866:
3819:
3747:
3691:
3627:
3607:
3544:
3524:
3494:
3474:
3422:
3369:
3303:
3283:
3237:
3217:
3187:
3108:
3088:
3068:
3005:
2983:
2963:
2937:
2894:
2870:
2800:
2772:
2745:
2703:
2683:
2663:
2643:
2611:
2583:
2563:
2539:
2519:
2495:
2471:
2447:
2427:
2399:
2272:
2229:
2186:
2152:
2132:
2112:
2092:
2056:
2027:
1999:
1979:
1951:
1931:
1911:
1863:
1837:
1811:
1779:
1749:
1726:
1706:
1679:
1651:
1631:
1611:
1591:
1541:
1507:
1481:
1461:
1415:
1387:
1367:
1344:
1324:
1279:
1259:
1232:
1212:
1192:
1172:
1081:
1024:
980:
951:
499:
474:
437:
1693:
is a lattice if and only if this measure is finite.
8099:. Ergebnisse de Mathematik und ihrer Grenzgebiete.
8038:. Ergebnisse de Mathematik und ihrer Grenzgebiete.
4486:There exists non-arithmetic lattices in all groups
8012:
7490:
7470:
7444:
7424:
7398:
7378:
7358:
7321:
7293:
7269:
7231:
7211:
7191:
7171:
7151:
7097:
7069:
7039:
6995:
6962:is a locally compact group (when endowed with the
6954:
6934:
6889:
6869:
6829:
6776:
6693:
6592:
6535:
6485:
6465:
6434:
6391:
6369:
6346:
6316:
6296:
6261:
6128:
6101:
6075:
6046:
6016:
5945:
5913:
5884:
5855:
5835:
5791:
5771:
5751:
5725:
5705:
5668:
5631:
5611:
5591:
5571:
5547:
5498:
5478:
5458:
5431:
5411:
5391:
5365:
5337:
5317:
5297:
5277:
5251:
5219:
5199:
5160:
5136:
5109:
5019:
4993:
4967:
4943:
4923:
4899:
4869:
4815:
4720:
4636:
4584:
4518:
4469:
4442:
4401:
4375:
4339:
4296:
4270:
4234:
4188:
4162:
4126:
4071:
4044:
4020:
3998:
3966:
3919:
3872:
3852:
3805:
3733:
3673:
3613:
3590:
3530:
3506:
3480:
3460:
3401:
3355:
3289:
3269:
3223:
3195:
3166:
3094:
3074:
3046:
2991:
2969:
2945:
2923:
2876:
2818:
2778:
2755:
2731:
2689:
2669:
2649:
2629:
2593:
2569:
2549:
2525:
2505:
2477:
2457:
2441:contains a lattice if and only if the Lie algebra
2433:
2405:
2330:
2258:
2215:
2166:
2138:
2118:
2098:
2071:
2042:
2005:
1985:
1965:
1937:
1917:
1897:
1849:
1823:
1793:
1761:
1732:
1712:
1685:
1665:
1637:
1617:
1597:
1574:
1527:
1493:
1467:
1447:
1401:
1373:
1350:
1330:
1310:
1265:
1245:
1218:
1198:
1178:
1139:
1054:
995:
966:
507:
482:
445:
4728:which have non-congruence finite-index subgroups.
3461:{\displaystyle G=G_{1}\times \ldots \times G_{r}}
6830:{\displaystyle \mathrm {PSL} _{2}(\mathbb {C} )}
6777:{\displaystyle \mathrm {PSL} _{2}(\mathbb {R} )}
3277:there is an obvious construction of lattices in
7897:Journal für die reine und angewandte Mathematik
7627:Gromov, Misha; Piatetski-Shapiro, Ilya (1987).
6694:{\displaystyle \mathrm {SL} _{2}(\mathbb {R} )}
6304:are noncompact then any irreducible lattice in
5684:of non-compact type without Euclidean factors.
3734:{\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )}
3102:; the simplest example of this is the subgroup
2924:{\displaystyle \mathrm {GL} _{n}(\mathbb {R} )}
2266:, and also by the higher-dimensional analogues
2259:{\displaystyle \mathrm {SL} _{2}(\mathbb {R} )}
2216:{\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )}
8019:. Progress in mathematics. Birkhäuser Verlag.
5399:is a Riemannian manifold locally isometric to
7679:Commensurabilities among Lattices in PU (1,n)
5285:are by definition isometries for this metric
4832:Lattices in semisimple Lie groups are always
1805:otherwise). Equivalently a discrete subgroup
809:
8:
7882:
7833:
7735:
7711:
7629:"Nonarithmetic groups in Lobachevsky spaces"
7239:is a tree lattice through its action on the
6435:{\displaystyle G=\mathrm {G} (\mathbb {A} )}
3413:if it does not come from this construction.
1305:
1292:
8036:Discrete subgroups of semisimple Lie groups
7614:
7602:
7590:
7578:
7566:
2854:Arithmetic groups and existence of lattices
2557:has only rational structure constants, and
2376:A stronger condition than unimodularity is
1448:{\displaystyle \mu (G/\Gamma )<+\infty }
5933:is at least 2 all irreducible lattices in
4637:{\displaystyle \mathrm {SU} (n,1),n\geq 4}
816:
802:
254:
80:
45:
8146:
8011:Bass, Hyman; Lubotzky, Alexander (2001).
7908:
7859:
7761:
7527:
7483:
7457:
7437:
7411:
7391:
7371:
7336:
7334:
7314:
7286:
7262:
7224:
7204:
7184:
7164:
7128:
7124:
7123:
7120:
7090:
7056:
7017:
7015:
6973:
6971:
6947:
6912:
6910:
6882:
6862:
6820:
6819:
6810:
6799:
6796:
6767:
6766:
6757:
6746:
6743:
6684:
6683:
6674:
6666:
6663:
6583:
6582:
6574:
6557:
6555:
6526:
6525:
6517:
6500:
6498:
6478:
6449:
6447:
6425:
6424:
6416:
6408:
6385:
6384:
6382:
6362:
6339:
6337:
6309:
6288:
6282:
6250:
6246:
6245:
6235:
6227:
6216:
6215:
6206:
6198:
6175:
6167:
6166:
6155:
6147:
6144:
6120:
6114:
6088:
6067:
6063:
6062:
6059:
6038:
6032:
6008:
5992:
5980:
5938:
5905:
5901:
5900:
5897:
5876:
5872:
5871:
5868:
5848:
5827:
5823:
5822:
5819:
5784:
5764:
5738:
5718:
5713:acts freely, properly discontinuously on
5692:
5658:
5647:
5624:
5604:
5584:
5564:
5539:
5538:
5536:
5491:
5471:
5451:
5424:
5404:
5378:
5358:
5330:
5310:
5290:
5264:
5235:
5212:
5185:
5173:
5153:
5128:
5122:
5095:
5079:
5066:
5038:
5032:
5006:
4980:
4960:
4936:
4915:
4914:
4912:
4891:
4885:
4862:
4790:
4764:
4762:
4695:
4669:
4667:
4599:
4597:
4559:
4533:
4531:
4493:
4491:
4461:
4455:
4431:
4426:
4420:
4388:
4356:
4314:
4312:
4283:
4251:
4209:
4207:
4175:
4143:
4101:
4099:
4061:
4037:
4014:
4013:
4011:
3991:
3957:
3951:
3938:
3932:
3910:
3904:
3891:
3885:
3865:
3818:
3796:
3795:
3786:
3778:
3767:
3766:
3757:
3749:
3746:
3718:
3711:
3710:
3701:
3693:
3690:
3663:
3658:
3637:
3632:
3626:
3606:
3580:
3575:
3554:
3549:
3543:
3523:
3493:
3473:
3452:
3433:
3421:
3387:
3374:
3368:
3347:
3334:
3321:
3308:
3302:
3282:
3261:
3248:
3236:
3216:
3189:
3188:
3186:
3157:
3156:
3147:
3139:
3128:
3127:
3118:
3110:
3107:
3087:
3067:
3037:
3036:
3027:
3019:
3004:
2985:
2984:
2982:
2962:
2939:
2938:
2936:
2914:
2913:
2904:
2896:
2893:
2869:
2799:
2771:
2747:
2746:
2744:
2708:
2702:
2682:
2662:
2642:
2610:
2585:
2584:
2582:
2562:
2541:
2540:
2538:
2518:
2513:are rational numbers. More precisely: if
2497:
2496:
2494:
2470:
2449:
2448:
2446:
2426:
2398:
2321:
2320:
2311:
2303:
2292:
2291:
2282:
2274:
2271:
2249:
2248:
2239:
2231:
2228:
2206:
2205:
2196:
2188:
2185:
2156:
2151:
2131:
2111:
2091:
2063:
2059:
2058:
2055:
2034:
2030:
2029:
2026:
1998:
1978:
1955:
1950:
1930:
1910:
1888:
1876:
1874:
1862:
1836:
1810:
1783:
1778:
1748:
1725:
1705:
1678:
1655:
1650:
1630:
1610:
1590:
1540:
1517:
1506:
1480:
1460:
1425:
1414:
1391:
1386:
1366:
1343:
1323:
1299:
1278:
1258:
1237:
1231:
1211:
1191:
1171:
1130:
1129:
1120:
1112:
1101:
1100:
1091:
1083:
1080:
1046:
1042:
1041:
1031:
1027:
1026:
1023:
987:
983:
982:
979:
958:
954:
953:
950:
501:
500:
498:
476:
475:
473:
439:
438:
436:
7801:Lubotzky, Alexander; Segal, Dan (2003).
7723:
7699:
7677:Deligne, Pierre; Mostow, George (1993).
7277:is the fundamental group of an infinite
2957:(i.e. the polynomial equations defining
2083:inside the continuous Heisenberg group.
33:
7506:
6354:is a semisimple algebraic group over a
5001:belong to the tangent space at a point
3853:{\displaystyle g\mapsto (g,\sigma (g))}
2146:of finite index (i.e. the quotient set
1773:(or cocompact) when the quotient space
1311:{\displaystyle \Gamma \cap U=\{e_{G}\}}
372:
138:
48:
6017:{\displaystyle G=\prod _{p\in S}G_{p}}
5200:{\displaystyle x\mapsto \gamma ^{-1}x}
3967:{\displaystyle a_{i}-b_{i}{\sqrt {2}}}
3920:{\displaystyle a_{i}+b_{i}{\sqrt {2}}}
2106:is a discrete group then a lattice in
852:. In the special case of subgroups of
374:Classification of finite simple groups
7452:be unimodular, and that the quotient
7219:-split rank one, then any lattice in
7152:{\displaystyle \mathbb {F} _{p}((t))}
6054:is a semisimple algebraic group over
4052:(an abelian subgroup containing only
7:
7247:Tree lattices from Bass–Serre theory
6604:. In particular modern forms of the
2050:which is a lattice in the Lie group
886:and automorphisms groups of regular
832:and related areas of mathematics, a
7081:Tree lattices from algebraic groups
7070:{\displaystyle \Gamma \backslash T}
5752:{\displaystyle \Gamma \backslash X}
5540:
5392:{\displaystyle \Gamma \backslash G}
4916:
4083:) those in the following list (see
3270:{\displaystyle G=G_{1}\times G_{2}}
2766:A lattice in a nilpotent Lie group
2748:
2732:{\displaystyle \exp ^{-1}(\Gamma )}
2586:
2542:
2498:
2450:
7406:being unimodular and the quotient
7343:
7340:
7337:
7288:
7264:
7058:
7024:
7021:
7018:
6980:
6977:
6974:
6919:
6916:
6913:
6806:
6803:
6800:
6753:
6750:
6747:
6670:
6667:
6575:
6558:
6518:
6501:
6450:
6417:
6340:
6231:
6228:
6202:
6199:
6151:
6148:
6121:
6096:
5740:
5694:
5493:
5380:
5312:
4794:
4791:
4768:
4765:
4699:
4696:
4673:
4670:
4603:
4600:
4563:
4560:
4537:
4534:
4519:{\displaystyle \mathrm {SO} (n,1)}
4497:
4494:
4340:{\displaystyle \mathrm {Sp} (n,1)}
4318:
4315:
4235:{\displaystyle \mathrm {SU} (n,1)}
4213:
4210:
4127:{\displaystyle \mathrm {SO} (n,1)}
4105:
4102:
3782:
3779:
3753:
3750:
3697:
3694:
3608:
3525:
3495:
3384:
3371:
3331:
3305:
3143:
3140:
3114:
3111:
3069:
3023:
3020:
3006:
2900:
2897:
2723:
2664:
2307:
2304:
2278:
2275:
2235:
2232:
2192:
2189:
2161:
2133:
1980:
1960:
1912:
1883:
1812:
1788:
1750:
1680:
1660:
1612:
1528:{\displaystyle W\subset G/\Gamma }
1522:
1442:
1430:
1396:
1325:
1280:
1193:
1116:
1113:
1087:
1084:
932:and other combinatorial objects).
917:(through the study of homogeneous
901:(as particularly nice examples of
14:
8119:Introduction to Arithmetic Groups
7750:Proc. Indian Acad. Sci. Math. Sci
7359:{\displaystyle \mathrm {Aut} (T)}
7040:{\displaystyle \mathrm {Aut} (T)}
6996:{\displaystyle \mathrm {Aut} (T)}
6935:{\displaystyle \mathrm {Aut} (T)}
6791:Nonuniform lattices in the group
5252:{\displaystyle x\mapsto \gamma x}
2849:Lattices in semisimple Lie groups
2601:(in the more elementary sense of
1475:-invariant (meaning that for any
8122:. Deductive Press. p. 492.
8097:Discrete subgroups of Lie groups
7179:an algebraic group defined over
6905:tree. The group of automorphisms
6076:{\displaystyle \mathbb {Q} _{p}}
5914:{\displaystyle \mathbb {Q} _{p}}
5885:{\displaystyle \mathbb {Q} _{p}}
5836:{\displaystyle \mathbb {Q} _{p}}
5706:{\displaystyle \Gamma \subset G}
4836:, and actually satisfy stronger
3507:{\displaystyle \Gamma \subset G}
2931:which is defined over the field
2072:{\displaystyle \mathbb {R} ^{n}}
2043:{\displaystyle \mathbb {Z} ^{n}}
1824:{\displaystyle \Gamma \subset G}
1762:{\displaystyle \Gamma \subset G}
1575:{\displaystyle \mu (gW)=\mu (W)}
996:{\displaystyle \mathbb {R} ^{n}}
967:{\displaystyle \mathbb {Z} ^{n}}
66:
6466:{\displaystyle \mathrm {G} (F)}
5548:{\displaystyle {\mathfrak {g}}}
4924:{\displaystyle {\mathfrak {g}}}
4662:but there are many lattices in
3488:into simple factors, a lattice
2756:{\displaystyle {\mathfrak {n}}}
2594:{\displaystyle {\mathfrak {n}}}
2550:{\displaystyle {\mathfrak {n}}}
2506:{\displaystyle {\mathfrak {n}}}
2458:{\displaystyle {\mathfrak {n}}}
2384:Lattices in solvable Lie groups
1186:be a locally compact group and
921:on the quotient spaces) and in
7681:. Princeton University Press.
7366:, under which conditions does
7353:
7347:
7146:
7143:
7137:
7134:
7034:
7028:
6990:
6984:
6929:
6923:
6824:
6816:
6771:
6763:
6688:
6680:
6587:
6579:
6568:
6562:
6530:
6522:
6511:
6505:
6460:
6454:
6429:
6421:
6256:
6241:
6220:
6212:
5240:
5178:
5104:
5072:
5056:
5044:
4810:
4798:
4784:
4772:
4715:
4703:
4689:
4677:
4619:
4607:
4579:
4567:
4553:
4541:
4513:
4501:
4370:
4358:
4334:
4322:
4265:
4253:
4229:
4217:
4157:
4145:
4121:
4109:
4006:is the maximal dimension of a
3847:
3844:
3838:
3826:
3823:
3800:
3792:
3771:
3763:
3728:
3725:
3715:
3707:
3161:
3153:
3132:
3124:
3041:
3033:
2918:
2910:
2813:
2807:
2726:
2720:
2624:
2618:
2325:
2317:
2296:
2288:
2253:
2245:
2210:
2202:
1993:is automatically a lattice in
1569:
1563:
1554:
1545:
1433:
1419:
1358:if in addition there exists a
1134:
1126:
1105:
1097:
735:Infinite dimensional Lie group
1:
7870:10.1215/S0012-7094-04-12432-7
7471:{\displaystyle H\backslash T}
7425:{\displaystyle H\backslash T}
6734:Nonrigidity in low dimensions
6726:into another algebraic group
5807:Lattices in p-adic Lie groups
925:(through the construction of
858:geometric notion of a lattice
8068:; Rapinchuk, Andrei (1994).
6548:strong approximation theorem
6392:{\displaystyle \mathbb {A} }
6347:{\displaystyle \mathrm {G} }
5325:is any discrete subgroup in
5278:{\displaystyle \gamma \in G}
5020:{\displaystyle \gamma \in G}
4877:is a Lie group then from an
4732:
4660:congruence subgroup property
4021:{\displaystyle \mathbb {R} }
3741:which we view as a subgroup
3297:from the smaller groups: if
3196:{\displaystyle \mathbb {Q} }
2992:{\displaystyle \mathbb {Q} }
2946:{\displaystyle \mathbb {Q} }
2358:is a well-understood topic.
1925:is any discrete subgroup in
1645:-invariant Borel measure on
1013:cardinality of the continuum
856:, this amounts to the usual
508:{\displaystyle \mathbb {Z} }
483:{\displaystyle \mathbb {Z} }
446:{\displaystyle \mathbb {Z} }
8116:Witte-Morris, Dave (2015).
8095:Raghunathan, M. S. (1972).
6129:{\displaystyle G_{\infty }}
5863:be an algebraic group over
5531:A natural bilinear form on
5137:{\displaystyle \gamma ^{*}}
4443:{\displaystyle F_{4}^{-20}}
3054:. A fundamental theorem of
2977:have their coefficients in
2349:Which groups have lattices?
2345:to a lattice is a lattice.
865:semisimple algebraic groups
844:with the property that the
233:List of group theory topics
8210:
8034:Margulis, Grigory (1991).
7250:
6109:is allowed, in which case
5524:
5466:defines a Haar measure on
4482:higher rank. For example:
2857:
1850:{\displaystyle C\subset G}
1605:is unimodular, then since
38:A portion of the discrete
19:For discrete subgroups of
18:
7848:Duke Mathematical Journal
6328:Lattices in adelic groups
6102:{\displaystyle p=\infty }
5639:-invariant metric on the
4347:(groups of matrices with
4085:List of simple Lie groups
3978:Rank 1 versus higher rank
2999:) then it has a subgroup
2367:upper triangular matrices
2167:{\displaystyle G/\Gamma }
1966:{\displaystyle G/\Gamma }
1794:{\displaystyle G/\Gamma }
1666:{\displaystyle G/\Gamma }
1402:{\displaystyle G/\Gamma }
1069:relatively compact subset
890:(the latter are known as
7329:is a closed subgroup of
6641:Kazhdan-Margulis theorem
6635:or by the topology on a
5521:Locally symmetric spaces
5353:by left-translations on
5351:properly discontinuously
5168:) of the diffeomorphism
3468:is the decomposition of
1226:of the identity element
936:Generalities on lattices
351:Elementary abelian group
228:Glossary of group theory
29:Lattice (disambiguation)
7919:10.1515/CRELLE.2011.085
7294:{\displaystyle \Gamma }
7270:{\displaystyle \Gamma }
6839:hyperbolic Dehn surgery
6656:Mostow rigidity theorem
5803:on the geometric side.
5499:{\displaystyle \Gamma }
5318:{\displaystyle \Gamma }
5161:{\displaystyle \gamma }
4402:{\displaystyle n\geq 2}
4297:{\displaystyle n\geq 2}
4189:{\displaystyle n\geq 2}
3873:{\displaystyle \sigma }
3614:{\displaystyle \Gamma }
3531:{\displaystyle \Gamma }
3082:is always a lattice in
3075:{\displaystyle \Gamma }
2819:{\displaystyle \dim(N)}
2739:generates a lattice in
2670:{\displaystyle \Gamma }
2637:generates a lattice in
2630:{\displaystyle \exp(L)}
2139:{\displaystyle \Gamma }
1986:{\displaystyle \Gamma }
1918:{\displaystyle \Gamma }
1686:{\displaystyle \Gamma }
1618:{\displaystyle \Gamma }
1338:is called a lattice in
1331:{\displaystyle \Gamma }
1199:{\displaystyle \Gamma }
8189:Geometric group theory
8139:Geometric group theory
7516:Bull. London Math. Soc
7492:
7472:
7446:
7426:
7400:
7380:
7360:
7323:
7295:
7271:
7233:
7213:
7193:
7173:
7153:
7099:
7071:
7041:
6997:
6956:
6936:
6891:
6871:
6831:
6778:
6695:
6594:
6537:
6487:
6467:
6436:
6393:
6371:
6348:
6318:
6298:
6263:
6130:
6103:
6077:
6048:
6018:
5947:
5915:
5886:
5857:
5837:
5793:
5773:
5753:
5727:
5707:
5670:
5633:
5613:
5593:
5573:
5549:
5500:
5480:
5460:
5444:Riemannian volume form
5433:
5413:
5393:
5367:
5339:
5319:
5299:
5279:
5253:
5221:
5201:
5162:
5138:
5111:
5021:
4995:
4969:
4951:) one can construct a
4945:
4925:
4901:
4871:
4853:Left-invariant metrics
4817:
4733:Kazhdan's property (T)
4722:
4638:
4586:
4520:
4471:
4444:
4403:
4377:
4341:
4298:
4272:
4236:
4190:
4164:
4128:
4073:
4046:
4022:
4000:
3968:
3921:
3874:
3854:
3807:
3735:
3675:
3615:
3592:
3532:
3508:
3482:
3462:
3403:
3357:
3291:
3271:
3225:
3197:
3168:
3096:
3076:
3048:
2993:
2971:
2947:
2925:
2886:linear algebraic group
2878:
2820:
2780:
2757:
2733:
2691:
2671:
2651:
2631:
2595:
2571:
2551:
2527:
2507:
2479:
2459:
2435:
2421:A nilpotent Lie group
2407:
2332:
2260:
2217:
2168:
2140:
2126:is exactly a subgroup
2120:
2100:
2073:
2044:
2007:
1987:
1967:
1939:
1919:
1899:
1851:
1825:
1795:
1763:
1734:
1714:
1687:
1667:
1639:
1619:
1599:
1576:
1529:
1495:
1494:{\displaystyle g\in G}
1469:
1449:
1409:which is finite (i.e.
1403:
1381:on the quotient space
1375:
1352:
1332:
1312:
1267:
1247:
1220:
1200:
1180:
1141:
1056:
997:
968:
899:geometric group theory
767:Linear algebraic group
509:
484:
447:
43:
27:. For other uses, see
8179:Differential geometry
7493:
7473:
7447:
7427:
7401:
7381:
7361:
7324:
7296:
7272:
7234:
7214:
7194:
7174:
7154:
7100:
7072:
7042:
6998:
6964:compact-open topology
6957:
6937:
6892:
6872:
6832:
6779:
6696:
6595:
6550:relates the quotient
6538:
6488:
6468:
6437:
6394:
6372:
6349:
6319:
6299:
6297:{\displaystyle G_{p}}
6264:
6131:
6104:
6078:
6049:
6047:{\displaystyle G_{p}}
6019:
5948:
5916:
5887:
5858:
5838:
5794:
5774:
5754:
5728:
5708:
5671:
5669:{\displaystyle X=G/K}
5634:
5614:
5594:
5574:
5550:
5501:
5481:
5461:
5434:
5414:
5394:
5368:
5340:
5320:
5300:
5280:
5254:
5222:
5202:
5163:
5139:
5112:
5022:
4996:
4970:
4946:
4926:
4907:on the tangent space
4902:
4900:{\displaystyle g_{e}}
4872:
4838:finiteness conditions
4828:Finiteness properties
4818:
4723:
4639:
4587:
4521:
4472:
4470:{\displaystyle F_{4}}
4445:
4414:exceptional Lie group
4404:
4378:
4376:{\displaystyle (n,1)}
4342:
4299:
4273:
4271:{\displaystyle (n,1)}
4237:
4191:
4165:
4163:{\displaystyle (n,1)}
4129:
4074:
4072:{\displaystyle \pm 1}
4047:
4023:
4001:
3969:
3922:
3875:
3855:
3808:
3736:
3676:
3616:
3593:
3533:
3509:
3483:
3463:
3404:
3358:
3292:
3272:
3226:
3198:
3169:
3097:
3077:
3049:
2994:
2972:
2948:
2926:
2879:
2821:
2781:
2758:
2734:
2692:
2672:
2652:
2632:
2596:
2572:
2552:
2528:
2508:
2480:
2460:
2436:
2408:
2333:
2261:
2218:
2169:
2141:
2121:
2101:
2074:
2045:
2008:
1988:
1968:
1940:
1920:
1900:
1852:
1826:
1796:
1764:
1735:
1715:
1688:
1668:
1640:
1620:
1600:
1577:
1530:
1496:
1470:
1450:
1404:
1376:
1353:
1333:
1313:
1268:
1248:
1246:{\displaystyle e_{G}}
1221:
1201:
1181:
1142:
1057:
998:
969:
907:differential geometry
838:locally compact group
510:
485:
448:
37:
8141:. pp. 249–282.
7975:Weil, André (1982).
7836:, Proposition 13.17.
7482:
7456:
7436:
7410:
7390:
7370:
7333:
7313:
7285:
7261:
7241:Bruhat–Tits building
7223:
7203:
7183:
7163:
7119:
7089:
7055:
7014:
6970:
6946:
6909:
6881:
6861:
6795:
6742:
6662:
6554:
6497:
6477:
6446:
6407:
6381:
6361:
6336:
6308:
6281:
6143:
6113:
6087:
6058:
6031:
5979:
5937:
5896:
5867:
5847:
5818:
5783:
5763:
5737:
5717:
5691:
5678:Riemannian manifolds
5646:
5623:
5603:
5583:
5563:
5535:
5490:
5470:
5450:
5423:
5403:
5377:
5357:
5329:
5309:
5305:. In particular, if
5289:
5263:
5234:
5211:
5172:
5152:
5121:
5031:
5005:
4979:
4959:
4935:
4931:(the Lie algebra of
4911:
4884:
4861:
4761:
4666:
4596:
4530:
4490:
4454:
4419:
4387:
4355:
4311:
4282:
4250:
4206:
4174:
4142:
4136:real quadratic forms
4098:
4060:
4036:
4010:
3990:
3931:
3884:
3864:
3817:
3745:
3689:
3625:
3605:
3601:The intersection of
3542:
3522:
3492:
3472:
3420:
3367:
3301:
3281:
3235:
3231:splits as a product
3215:
3185:
3106:
3086:
3066:
3003:
2981:
2961:
2935:
2892:
2868:
2798:
2770:
2743:
2701:
2681:
2661:
2641:
2609:
2581:
2561:
2537:
2517:
2493:
2469:
2445:
2425:
2397:
2389:Nilpotent Lie groups
2270:
2227:
2184:
2150:
2130:
2110:
2090:
2054:
2025:
1997:
1977:
1949:
1929:
1909:
1861:
1835:
1809:
1777:
1747:
1724:
1704:
1677:
1673:up to scaling. Then
1649:
1629:
1609:
1589:
1539:
1505:
1501:and any open subset
1479:
1459:
1413:
1385:
1374:{\displaystyle \mu }
1365:
1342:
1322:
1277:
1257:
1230:
1210:
1190:
1170:
1079:
1022:
978:
949:
497:
472:
435:
8157:2014arXiv1402.0962G
7726:, pp. 263–270.
7538:10.1112/blms/bdr061
7305:Existence criterion
5968:S-arithmetic groups
4994:{\displaystyle v,w}
4439:
3211:When the Lie group
2487:structure constants
941:Informal discussion
141:Group homomorphisms
51:Algebraic structure
8066:Platonov, Vladimir
8042:. pp. x+388.
7946:10.1007/BF01895641
7772:10.1007/BF02829437
7648:10.1007/bf02698928
7488:
7468:
7442:
7422:
7396:
7376:
7356:
7319:
7291:
7267:
7229:
7209:
7189:
7169:
7149:
7095:
7067:
7037:
6993:
6952:
6932:
6887:
6867:
6827:
6786:Teichmüller spaces
6774:
6691:
6590:
6533:
6483:
6463:
6432:
6389:
6367:
6344:
6314:
6294:
6275:S-arithmetic group
6259:
6126:
6099:
6073:
6044:
6014:
6003:
5943:
5911:
5882:
5853:
5833:
5789:
5769:
5749:
5723:
5703:
5666:
5629:
5609:
5599:is semisimple and
5589:
5569:
5545:
5496:
5476:
5456:
5429:
5409:
5389:
5363:
5335:
5315:
5295:
5275:
5249:
5217:
5197:
5158:
5134:
5107:
5017:
4991:
4965:
4941:
4921:
4897:
4867:
4834:finitely presented
4813:
4718:
4634:
4592:, and possibly in
4582:
4516:
4467:
4440:
4422:
4399:
4373:
4337:
4294:
4268:
4232:
4186:
4160:
4124:
4069:
4042:
4018:
3996:
3964:
3917:
3870:
3850:
3803:
3731:
3671:
3611:
3588:
3528:
3518:The projection of
3504:
3478:
3458:
3416:More formally, if
3399:
3363:are lattices then
3353:
3287:
3267:
3221:
3193:
3179:arithmetic lattice
3164:
3092:
3072:
3044:
2989:
2967:
2943:
2921:
2874:
2816:
2792:finitely presented
2788:finitely generated
2776:
2753:
2729:
2687:
2667:
2647:
2627:
2591:
2567:
2547:
2523:
2503:
2475:
2455:
2431:
2403:
2328:
2256:
2213:
2164:
2136:
2116:
2096:
2069:
2040:
2003:
1983:
1963:
1935:
1915:
1895:
1847:
1821:
1791:
1759:
1730:
1710:
1683:
1663:
1635:
1615:
1595:
1572:
1525:
1491:
1465:
1445:
1399:
1371:
1348:
1328:
1308:
1263:
1243:
1216:
1196:
1176:
1152:coarse equivalence
1150:Other notions are
1137:
1052:
1005:finitely generated
993:
964:
884:Kac–Moody algebras
617:Special orthogonal
505:
480:
443:
324:Lagrange's theorem
44:
8129:978-0-9865716-0-2
7934:Geom. Funct. Anal
7883:Witte-Morris 2015
7834:Witte-Morris 2015
7736:Witte-Morris 2015
7712:Witte-Morris 2015
7491:{\displaystyle H}
7445:{\displaystyle H}
7399:{\displaystyle H}
7379:{\displaystyle H}
7322:{\displaystyle H}
7253:Bass–Serre theory
7232:{\displaystyle G}
7212:{\displaystyle F}
7192:{\displaystyle F}
7172:{\displaystyle G}
7098:{\displaystyle F}
7003:is then called a
6955:{\displaystyle T}
6890:{\displaystyle T}
6870:{\displaystyle T}
6637:character variety
6633:Chabauty topology
6602:automorphic forms
6486:{\displaystyle F}
6370:{\displaystyle F}
6324:is S-arithmetic.
6317:{\displaystyle G}
6183:
5988:
5946:{\displaystyle G}
5856:{\displaystyle G}
5792:{\displaystyle G}
5772:{\displaystyle X}
5726:{\displaystyle X}
5641:homogeneous space
5632:{\displaystyle G}
5612:{\displaystyle K}
5592:{\displaystyle G}
5572:{\displaystyle G}
5479:{\displaystyle G}
5459:{\displaystyle g}
5432:{\displaystyle g}
5412:{\displaystyle G}
5366:{\displaystyle G}
5345:(so that it acts
5338:{\displaystyle G}
5298:{\displaystyle g}
5220:{\displaystyle G}
4968:{\displaystyle G}
4953:Riemannian metric
4944:{\displaystyle G}
4870:{\displaystyle G}
4750:harmonic analysis
4093:orthogonal groups
4045:{\displaystyle G}
3999:{\displaystyle G}
3962:
3915:
3723:
3681:is not a lattice.
3481:{\displaystyle G}
3290:{\displaystyle G}
3224:{\displaystyle G}
3095:{\displaystyle G}
2970:{\displaystyle G}
2877:{\displaystyle G}
2779:{\displaystyle N}
2690:{\displaystyle N}
2657:; conversely, if
2650:{\displaystyle N}
2570:{\displaystyle L}
2526:{\displaystyle N}
2478:{\displaystyle N}
2434:{\displaystyle N}
2406:{\displaystyle N}
2119:{\displaystyle G}
2099:{\displaystyle G}
2006:{\displaystyle G}
1938:{\displaystyle G}
1733:{\displaystyle G}
1713:{\displaystyle G}
1638:{\displaystyle G}
1598:{\displaystyle G}
1468:{\displaystyle G}
1351:{\displaystyle G}
1266:{\displaystyle G}
1219:{\displaystyle U}
1179:{\displaystyle G}
1154:and the stronger
911:arithmetic groups
877:arithmetic groups
850:invariant measure
842:discrete subgroup
826:
825:
401:
400:
283:Alternating group
240:
239:
8201:
8174:Algebraic groups
8160:
8150:
8133:
8112:
8091:
8061:
8030:
8018:
7999:
7998:
7972:
7966:
7965:
7929:
7923:
7922:
7912:
7903:(661): 237–248.
7892:
7886:
7880:
7874:
7873:
7863:
7843:
7837:
7831:
7825:
7824:
7798:
7792:
7791:
7765:
7745:
7739:
7733:
7727:
7721:
7715:
7709:
7703:
7697:
7691:
7690:
7674:
7668:
7667:
7636:Publ. Math. IHÉS
7633:
7624:
7618:
7615:Raghunathan 1972
7612:
7606:
7603:Raghunathan 1972
7600:
7594:
7591:Raghunathan 1972
7588:
7582:
7579:Raghunathan 1972
7576:
7570:
7567:Raghunathan 1972
7564:
7558:
7557:
7531:
7511:
7497:
7495:
7494:
7489:
7477:
7475:
7474:
7469:
7451:
7449:
7448:
7443:
7431:
7429:
7428:
7423:
7405:
7403:
7402:
7397:
7385:
7383:
7382:
7377:
7365:
7363:
7362:
7357:
7346:
7328:
7326:
7325:
7320:
7300:
7298:
7297:
7292:
7276:
7274:
7273:
7268:
7238:
7236:
7235:
7230:
7218:
7216:
7215:
7210:
7198:
7196:
7195:
7190:
7178:
7176:
7175:
7170:
7158:
7156:
7155:
7150:
7133:
7132:
7127:
7104:
7102:
7101:
7096:
7076:
7074:
7073:
7068:
7046:
7044:
7043:
7038:
7027:
7002:
7000:
6999:
6994:
6983:
6961:
6959:
6958:
6953:
6941:
6939:
6938:
6933:
6922:
6896:
6894:
6893:
6888:
6876:
6874:
6873:
6868:
6836:
6834:
6833:
6828:
6823:
6815:
6814:
6809:
6783:
6781:
6780:
6775:
6770:
6762:
6761:
6756:
6700:
6698:
6697:
6692:
6687:
6679:
6678:
6673:
6617:Rigidity results
6599:
6597:
6596:
6591:
6586:
6578:
6561:
6542:
6540:
6539:
6534:
6529:
6521:
6504:
6492:
6490:
6489:
6484:
6472:
6470:
6469:
6464:
6453:
6441:
6439:
6438:
6433:
6428:
6420:
6398:
6396:
6395:
6390:
6388:
6376:
6374:
6373:
6368:
6353:
6351:
6350:
6345:
6343:
6323:
6321:
6320:
6315:
6303:
6301:
6300:
6295:
6293:
6292:
6268:
6266:
6265:
6260:
6255:
6254:
6249:
6240:
6239:
6234:
6219:
6211:
6210:
6205:
6193:
6189:
6188:
6184:
6176:
6170:
6160:
6159:
6154:
6135:
6133:
6132:
6127:
6125:
6124:
6108:
6106:
6105:
6100:
6082:
6080:
6079:
6074:
6072:
6071:
6066:
6053:
6051:
6050:
6045:
6043:
6042:
6023:
6021:
6020:
6015:
6013:
6012:
6002:
5952:
5950:
5949:
5944:
5920:
5918:
5917:
5912:
5910:
5909:
5904:
5891:
5889:
5888:
5883:
5881:
5880:
5875:
5862:
5860:
5859:
5854:
5842:
5840:
5839:
5834:
5832:
5831:
5826:
5798:
5796:
5795:
5790:
5778:
5776:
5775:
5770:
5758:
5756:
5755:
5750:
5732:
5730:
5729:
5724:
5712:
5710:
5709:
5704:
5682:symmetric spaces
5675:
5673:
5672:
5667:
5662:
5638:
5636:
5635:
5630:
5618:
5616:
5615:
5610:
5598:
5596:
5595:
5590:
5578:
5576:
5575:
5570:
5555:is given by the
5554:
5552:
5551:
5546:
5544:
5543:
5505:
5503:
5502:
5497:
5485:
5483:
5482:
5477:
5465:
5463:
5462:
5457:
5438:
5436:
5435:
5430:
5419:with the metric
5418:
5416:
5415:
5410:
5398:
5396:
5395:
5390:
5372:
5370:
5369:
5364:
5344:
5342:
5341:
5336:
5324:
5322:
5321:
5316:
5304:
5302:
5301:
5296:
5284:
5282:
5281:
5276:
5258:
5256:
5255:
5250:
5226:
5224:
5223:
5218:
5206:
5204:
5203:
5198:
5193:
5192:
5167:
5165:
5164:
5159:
5143:
5141:
5140:
5135:
5133:
5132:
5116:
5114:
5113:
5108:
5100:
5099:
5084:
5083:
5071:
5070:
5043:
5042:
5026:
5024:
5023:
5018:
5000:
4998:
4997:
4992:
4974:
4972:
4971:
4966:
4950:
4948:
4947:
4942:
4930:
4928:
4927:
4922:
4920:
4919:
4906:
4904:
4903:
4898:
4896:
4895:
4876:
4874:
4873:
4868:
4822:
4820:
4819:
4814:
4797:
4771:
4727:
4725:
4724:
4719:
4702:
4676:
4653:normal subgroups
4644:(the last is an
4643:
4641:
4640:
4635:
4606:
4591:
4589:
4588:
4583:
4566:
4540:
4525:
4523:
4522:
4517:
4500:
4476:
4474:
4473:
4468:
4466:
4465:
4449:
4447:
4446:
4441:
4438:
4430:
4408:
4406:
4405:
4400:
4382:
4380:
4379:
4374:
4346:
4344:
4343:
4338:
4321:
4303:
4301:
4300:
4295:
4277:
4275:
4274:
4269:
4241:
4239:
4238:
4233:
4216:
4195:
4193:
4192:
4187:
4169:
4167:
4166:
4161:
4133:
4131:
4130:
4125:
4108:
4078:
4076:
4075:
4070:
4051:
4049:
4048:
4043:
4027:
4025:
4024:
4019:
4017:
4005:
4003:
4002:
3997:
3973:
3971:
3970:
3965:
3963:
3958:
3956:
3955:
3943:
3942:
3926:
3924:
3923:
3918:
3916:
3911:
3909:
3908:
3896:
3895:
3879:
3877:
3876:
3871:
3859:
3857:
3856:
3851:
3812:
3810:
3809:
3804:
3799:
3791:
3790:
3785:
3770:
3762:
3761:
3756:
3740:
3738:
3737:
3732:
3724:
3719:
3714:
3706:
3705:
3700:
3680:
3678:
3677:
3672:
3670:
3669:
3668:
3667:
3644:
3643:
3642:
3641:
3621:with any factor
3620:
3618:
3617:
3612:
3597:
3595:
3594:
3589:
3587:
3586:
3585:
3584:
3561:
3560:
3559:
3558:
3537:
3535:
3534:
3529:
3513:
3511:
3510:
3505:
3487:
3485:
3484:
3479:
3467:
3465:
3464:
3459:
3457:
3456:
3438:
3437:
3408:
3406:
3405:
3400:
3392:
3391:
3379:
3378:
3362:
3360:
3359:
3354:
3352:
3351:
3339:
3338:
3326:
3325:
3313:
3312:
3296:
3294:
3293:
3288:
3276:
3274:
3273:
3268:
3266:
3265:
3253:
3252:
3230:
3228:
3227:
3222:
3202:
3200:
3199:
3194:
3192:
3173:
3171:
3170:
3165:
3160:
3152:
3151:
3146:
3131:
3123:
3122:
3117:
3101:
3099:
3098:
3093:
3081:
3079:
3078:
3073:
3053:
3051:
3050:
3045:
3040:
3032:
3031:
3026:
2998:
2996:
2995:
2990:
2988:
2976:
2974:
2973:
2968:
2955:rational numbers
2952:
2950:
2949:
2944:
2942:
2930:
2928:
2927:
2922:
2917:
2909:
2908:
2903:
2884:is a semisimple
2883:
2881:
2880:
2875:
2860:Arithmetic group
2833:The general case
2825:
2823:
2822:
2817:
2785:
2783:
2782:
2777:
2762:
2760:
2759:
2754:
2752:
2751:
2738:
2736:
2735:
2730:
2716:
2715:
2696:
2694:
2693:
2688:
2677:is a lattice in
2676:
2674:
2673:
2668:
2656:
2654:
2653:
2648:
2636:
2634:
2633:
2628:
2600:
2598:
2597:
2592:
2590:
2589:
2577:is a lattice in
2576:
2574:
2573:
2568:
2556:
2554:
2553:
2548:
2546:
2545:
2532:
2530:
2529:
2524:
2512:
2510:
2509:
2504:
2502:
2501:
2484:
2482:
2481:
2476:
2464:
2462:
2461:
2456:
2454:
2453:
2440:
2438:
2437:
2432:
2415:simply connected
2412:
2410:
2409:
2404:
2337:
2335:
2334:
2329:
2324:
2316:
2315:
2310:
2295:
2287:
2286:
2281:
2265:
2263:
2262:
2257:
2252:
2244:
2243:
2238:
2222:
2220:
2219:
2214:
2209:
2201:
2200:
2195:
2173:
2171:
2170:
2165:
2160:
2145:
2143:
2142:
2137:
2125:
2123:
2122:
2117:
2105:
2103:
2102:
2097:
2081:Heisenberg group
2078:
2076:
2075:
2070:
2068:
2067:
2062:
2049:
2047:
2046:
2041:
2039:
2038:
2033:
2012:
2010:
2009:
2004:
1992:
1990:
1989:
1984:
1973:is compact then
1972:
1970:
1969:
1964:
1959:
1944:
1942:
1941:
1936:
1924:
1922:
1921:
1916:
1904:
1902:
1901:
1896:
1887:
1886:
1875:
1856:
1854:
1853:
1848:
1830:
1828:
1827:
1822:
1801:is compact (and
1800:
1798:
1797:
1792:
1787:
1768:
1766:
1765:
1760:
1739:
1737:
1736:
1731:
1719:
1717:
1716:
1711:
1692:
1690:
1689:
1684:
1672:
1670:
1669:
1664:
1659:
1644:
1642:
1641:
1636:
1624:
1622:
1621:
1616:
1604:
1602:
1601:
1596:
1581:
1579:
1578:
1573:
1534:
1532:
1531:
1526:
1521:
1500:
1498:
1497:
1492:
1474:
1472:
1471:
1466:
1454:
1452:
1451:
1446:
1429:
1408:
1406:
1405:
1400:
1395:
1380:
1378:
1377:
1372:
1357:
1355:
1354:
1349:
1337:
1335:
1334:
1329:
1317:
1315:
1314:
1309:
1304:
1303:
1272:
1270:
1269:
1264:
1252:
1250:
1249:
1244:
1242:
1241:
1225:
1223:
1222:
1217:
1205:
1203:
1202:
1197:
1185:
1183:
1182:
1177:
1146:
1144:
1143:
1138:
1133:
1125:
1124:
1119:
1104:
1096:
1095:
1090:
1061:
1059:
1058:
1053:
1051:
1050:
1045:
1036:
1035:
1030:
1002:
1000:
999:
994:
992:
991:
986:
973:
971:
970:
965:
963:
962:
957:
873:Grigory Margulis
818:
811:
804:
760:Algebraic groups
533:Hyperbolic group
523:Arithmetic group
514:
512:
511:
506:
504:
489:
487:
486:
481:
479:
452:
450:
449:
444:
442:
365:Schur multiplier
319:Cauchy's theorem
307:Quaternion group
255:
81:
70:
57:
46:
40:Heisenberg group
8209:
8208:
8204:
8203:
8202:
8200:
8199:
8198:
8164:
8163:
8136:
8130:
8115:
8101:Springer-Verlag
8094:
8080:
8064:
8050:
8040:Springer-Verlag
8033:
8027:
8010:
8007:
8002:
7987:
7974:
7973:
7969:
7931:
7930:
7926:
7894:
7893:
7889:
7881:
7877:
7845:
7844:
7840:
7832:
7828:
7813:
7803:Subgroup growth
7800:
7799:
7795:
7747:
7746:
7742:
7738:, Theorem 17.1.
7734:
7730:
7722:
7718:
7714:, Theorem 5.21.
7710:
7706:
7698:
7694:
7676:
7675:
7671:
7631:
7626:
7625:
7621:
7617:, Theorem 4.28.
7613:
7609:
7601:
7597:
7593:, Theorem 2.21.
7589:
7585:
7581:, Theorem 2.12.
7577:
7573:
7565:
7561:
7513:
7512:
7508:
7504:
7480:
7479:
7454:
7453:
7434:
7433:
7408:
7407:
7388:
7387:
7368:
7367:
7331:
7330:
7311:
7310:
7307:
7283:
7282:
7279:graph of groups
7259:
7258:
7255:
7249:
7221:
7220:
7201:
7200:
7181:
7180:
7161:
7160:
7122:
7117:
7116:
7087:
7086:
7083:
7053:
7052:
7012:
7011:
6968:
6967:
6944:
6943:
6907:
6906:
6879:
6878:
6859:
6858:
6855:
6850:
6798:
6793:
6792:
6745:
6740:
6739:
6736:
6703:strong rigidity
6665:
6660:
6659:
6619:
6614:
6552:
6551:
6495:
6494:
6475:
6474:
6444:
6443:
6405:
6404:
6403:then the group
6379:
6378:
6359:
6358:
6334:
6333:
6330:
6306:
6305:
6284:
6279:
6278:
6244:
6226:
6197:
6171:
6165:
6161:
6146:
6141:
6140:
6116:
6111:
6110:
6085:
6084:
6061:
6056:
6055:
6034:
6029:
6028:
6004:
5977:
5976:
5970:
5953:are arithmetic;
5935:
5934:
5899:
5894:
5893:
5870:
5865:
5864:
5845:
5844:
5821:
5816:
5815:
5809:
5781:
5780:
5761:
5760:
5735:
5734:
5715:
5714:
5689:
5688:
5644:
5643:
5621:
5620:
5601:
5600:
5581:
5580:
5561:
5560:
5533:
5532:
5529:
5527:Symmetric space
5523:
5488:
5487:
5468:
5467:
5448:
5447:
5421:
5420:
5401:
5400:
5375:
5374:
5373:) the quotient
5355:
5354:
5327:
5326:
5307:
5306:
5287:
5286:
5261:
5260:
5232:
5231:
5209:
5208:
5181:
5170:
5169:
5150:
5149:
5124:
5119:
5118:
5091:
5075:
5062:
5034:
5029:
5028:
5003:
5002:
4977:
4976:
4975:as follows: if
4957:
4956:
4933:
4932:
4909:
4908:
4887:
4882:
4881:
4859:
4858:
4855:
4850:
4830:
4759:
4758:
4735:
4664:
4663:
4594:
4593:
4528:
4527:
4488:
4487:
4457:
4452:
4451:
4417:
4416:
4385:
4384:
4353:
4352:
4309:
4308:
4280:
4279:
4248:
4247:
4244:Hermitian forms
4204:
4203:
4172:
4171:
4140:
4139:
4096:
4095:
4058:
4057:
4034:
4033:
4008:
4007:
3988:
3987:
3986:of a Lie group
3980:
3947:
3934:
3929:
3928:
3900:
3887:
3882:
3881:
3862:
3861:
3815:
3814:
3777:
3748:
3743:
3742:
3692:
3687:
3686:
3659:
3654:
3633:
3628:
3623:
3622:
3603:
3602:
3576:
3571:
3550:
3545:
3540:
3539:
3520:
3519:
3490:
3489:
3470:
3469:
3448:
3429:
3418:
3417:
3383:
3370:
3365:
3364:
3343:
3330:
3317:
3304:
3299:
3298:
3279:
3278:
3257:
3244:
3233:
3232:
3213:
3212:
3209:
3183:
3182:
3138:
3109:
3104:
3103:
3084:
3083:
3064:
3063:
3018:
3001:
3000:
2979:
2978:
2959:
2958:
2933:
2932:
2895:
2890:
2889:
2866:
2865:
2862:
2856:
2851:
2835:
2796:
2795:
2768:
2767:
2741:
2740:
2704:
2699:
2698:
2679:
2678:
2659:
2658:
2639:
2638:
2607:
2606:
2603:Lattice (group)
2579:
2578:
2559:
2558:
2535:
2534:
2515:
2514:
2491:
2490:
2467:
2466:
2443:
2442:
2423:
2422:
2418:vector space).
2413:is a connected
2395:
2394:
2391:
2386:
2351:
2302:
2273:
2268:
2267:
2230:
2225:
2224:
2187:
2182:
2181:
2148:
2147:
2128:
2127:
2108:
2107:
2088:
2087:
2057:
2052:
2051:
2028:
2023:
2022:
2019:
1995:
1994:
1975:
1974:
1947:
1946:
1927:
1926:
1907:
1906:
1905:. Note that if
1873:
1859:
1858:
1833:
1832:
1807:
1806:
1775:
1774:
1745:
1744:
1722:
1721:
1702:
1701:
1675:
1674:
1647:
1646:
1627:
1626:
1607:
1606:
1587:
1586:
1582:is satisfied).
1537:
1536:
1503:
1502:
1477:
1476:
1457:
1456:
1411:
1410:
1383:
1382:
1363:
1362:
1340:
1339:
1320:
1319:
1295:
1275:
1274:
1255:
1254:
1233:
1228:
1227:
1208:
1207:
1188:
1187:
1168:
1167:
1164:
1111:
1082:
1077:
1076:
1040:
1025:
1020:
1019:
981:
976:
975:
952:
947:
946:
943:
938:
903:discrete groups
822:
793:
792:
781:Abelian variety
774:Reductive group
762:
752:
751:
750:
749:
700:
692:
684:
676:
668:
641:Special unitary
552:
538:
537:
519:
518:
495:
494:
470:
469:
433:
432:
424:
423:
414:Discrete groups
403:
402:
358:Frobenius group
303:
290:
279:
272:Symmetric group
268:
252:
242:
241:
92:Normal subgroup
78:
58:
49:
32:
25:Lattice (group)
17:
12:
11:
5:
8207:
8205:
8197:
8196:
8191:
8186:
8184:Ergodic theory
8181:
8176:
8166:
8165:
8162:
8161:
8134:
8128:
8113:
8092:
8078:
8062:
8048:
8031:
8025:
8006:
8003:
8001:
8000:
7985:
7967:
7940:(4): 406–431.
7924:
7887:
7875:
7854:(3): 459–515.
7838:
7826:
7811:
7793:
7756:(4): 299–308.
7740:
7728:
7716:
7704:
7702:, p. 298.
7692:
7669:
7619:
7607:
7605:, Theorem 3.1.
7595:
7583:
7571:
7569:, Theorem 2.1.
7559:
7505:
7503:
7500:
7487:
7467:
7464:
7461:
7441:
7421:
7418:
7415:
7395:
7375:
7355:
7352:
7349:
7345:
7342:
7339:
7318:
7306:
7303:
7290:
7266:
7251:Main article:
7248:
7245:
7228:
7208:
7188:
7168:
7148:
7145:
7142:
7139:
7136:
7131:
7126:
7107:function field
7094:
7082:
7079:
7066:
7063:
7060:
7036:
7033:
7030:
7026:
7023:
7020:
6992:
6989:
6986:
6982:
6979:
6976:
6951:
6931:
6928:
6925:
6921:
6918:
6915:
6886:
6866:
6854:
6851:
6849:
6846:
6826:
6822:
6818:
6813:
6808:
6805:
6802:
6773:
6769:
6765:
6760:
6755:
6752:
6749:
6735:
6732:
6705:and is due to
6690:
6686:
6682:
6677:
6672:
6669:
6629:Local rigidity
6618:
6615:
6613:
6610:
6589:
6585:
6581:
6577:
6573:
6570:
6567:
6564:
6560:
6543:is a lattice.
6532:
6528:
6524:
6520:
6516:
6513:
6510:
6507:
6503:
6482:
6462:
6459:
6456:
6452:
6431:
6427:
6423:
6419:
6415:
6412:
6387:
6366:
6342:
6329:
6326:
6313:
6291:
6287:
6271:
6270:
6258:
6253:
6248:
6243:
6238:
6233:
6230:
6225:
6222:
6218:
6214:
6209:
6204:
6201:
6196:
6192:
6187:
6182:
6179:
6174:
6169:
6164:
6158:
6153:
6150:
6123:
6119:
6098:
6095:
6092:
6070:
6065:
6041:
6037:
6025:
6024:
6011:
6007:
6001:
5998:
5995:
5991:
5987:
5984:
5969:
5966:
5962:
5961:
5954:
5942:
5908:
5903:
5879:
5874:
5852:
5830:
5825:
5808:
5805:
5788:
5768:
5748:
5745:
5742:
5722:
5702:
5699:
5696:
5665:
5661:
5657:
5654:
5651:
5628:
5608:
5588:
5568:
5542:
5525:Main article:
5522:
5519:
5511:flat manifolds
5506:is a lattice.
5495:
5475:
5455:
5446:associated to
5428:
5408:
5388:
5385:
5382:
5362:
5334:
5314:
5294:
5274:
5271:
5268:
5248:
5245:
5242:
5239:
5216:
5196:
5191:
5188:
5184:
5180:
5177:
5157:
5144:indicates the
5131:
5127:
5106:
5103:
5098:
5094:
5090:
5087:
5082:
5078:
5074:
5069:
5065:
5061:
5058:
5055:
5052:
5049:
5046:
5041:
5037:
5016:
5013:
5010:
4990:
4987:
4984:
4964:
4940:
4918:
4894:
4890:
4866:
4854:
4851:
4849:
4846:
4829:
4826:
4825:
4824:
4812:
4809:
4806:
4803:
4800:
4796:
4793:
4789:
4786:
4783:
4780:
4777:
4774:
4770:
4767:
4746:
4745:
4734:
4731:
4730:
4729:
4717:
4714:
4711:
4708:
4705:
4701:
4698:
4694:
4691:
4688:
4685:
4682:
4679:
4675:
4672:
4656:
4649:
4633:
4630:
4627:
4624:
4621:
4618:
4615:
4612:
4609:
4605:
4602:
4581:
4578:
4575:
4572:
4569:
4565:
4562:
4558:
4555:
4552:
4549:
4546:
4543:
4539:
4536:
4515:
4512:
4509:
4506:
4503:
4499:
4496:
4479:
4478:
4464:
4460:
4437:
4434:
4429:
4425:
4410:
4398:
4395:
4392:
4372:
4369:
4366:
4363:
4360:
4336:
4333:
4330:
4327:
4324:
4320:
4317:
4305:
4293:
4290:
4287:
4267:
4264:
4261:
4258:
4255:
4231:
4228:
4225:
4222:
4219:
4215:
4212:
4201:unitary groups
4197:
4185:
4182:
4179:
4159:
4156:
4153:
4150:
4147:
4123:
4120:
4117:
4114:
4111:
4107:
4104:
4068:
4065:
4041:
4016:
3995:
3979:
3976:
3961:
3954:
3950:
3946:
3941:
3937:
3914:
3907:
3903:
3899:
3894:
3890:
3869:
3849:
3846:
3843:
3840:
3837:
3834:
3831:
3828:
3825:
3822:
3802:
3798:
3794:
3789:
3784:
3781:
3776:
3773:
3769:
3765:
3760:
3755:
3752:
3730:
3727:
3722:
3717:
3713:
3709:
3704:
3699:
3696:
3683:
3682:
3666:
3662:
3657:
3653:
3650:
3647:
3640:
3636:
3631:
3610:
3599:
3583:
3579:
3574:
3570:
3567:
3564:
3557:
3553:
3548:
3538:to any factor
3527:
3503:
3500:
3497:
3477:
3455:
3451:
3447:
3444:
3441:
3436:
3432:
3428:
3425:
3398:
3395:
3390:
3386:
3382:
3377:
3373:
3350:
3346:
3342:
3337:
3333:
3329:
3324:
3320:
3316:
3311:
3307:
3286:
3264:
3260:
3256:
3251:
3247:
3243:
3240:
3220:
3208:
3207:Irreducibility
3205:
3191:
3163:
3159:
3155:
3150:
3145:
3142:
3137:
3134:
3130:
3126:
3121:
3116:
3113:
3091:
3071:
3060:Harish-Chandra
3043:
3039:
3035:
3030:
3025:
3022:
3017:
3014:
3011:
3008:
2987:
2966:
2941:
2920:
2916:
2912:
2907:
2902:
2899:
2873:
2858:Main article:
2855:
2852:
2850:
2847:
2834:
2831:
2815:
2812:
2809:
2806:
2803:
2775:
2750:
2728:
2725:
2722:
2719:
2714:
2711:
2707:
2686:
2666:
2646:
2626:
2623:
2620:
2617:
2614:
2588:
2566:
2544:
2522:
2500:
2474:
2452:
2430:
2402:
2390:
2387:
2385:
2382:
2350:
2347:
2327:
2323:
2319:
2314:
2309:
2306:
2301:
2298:
2294:
2290:
2285:
2280:
2277:
2255:
2251:
2247:
2242:
2237:
2234:
2212:
2208:
2204:
2199:
2194:
2191:
2163:
2159:
2155:
2135:
2115:
2095:
2066:
2061:
2037:
2032:
2018:
2017:First examples
2015:
2002:
1982:
1962:
1958:
1954:
1934:
1914:
1894:
1891:
1885:
1882:
1879:
1872:
1869:
1866:
1846:
1843:
1840:
1820:
1817:
1814:
1790:
1786:
1782:
1758:
1755:
1752:
1729:
1709:
1682:
1662:
1658:
1654:
1634:
1614:
1594:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1547:
1544:
1524:
1520:
1516:
1513:
1510:
1490:
1487:
1484:
1464:
1444:
1441:
1438:
1435:
1432:
1428:
1424:
1421:
1418:
1398:
1394:
1390:
1370:
1347:
1327:
1307:
1302:
1298:
1294:
1291:
1288:
1285:
1282:
1262:
1240:
1236:
1215:
1195:
1175:
1163:
1160:
1156:quasi-isometry
1136:
1132:
1128:
1123:
1118:
1115:
1110:
1107:
1103:
1099:
1094:
1089:
1086:
1049:
1044:
1039:
1034:
1029:
990:
985:
961:
956:
942:
939:
937:
934:
915:ergodic theory
846:quotient space
824:
823:
821:
820:
813:
806:
798:
795:
794:
791:
790:
788:Elliptic curve
784:
783:
777:
776:
770:
769:
763:
758:
757:
754:
753:
748:
747:
744:
741:
737:
733:
732:
731:
726:
724:Diffeomorphism
720:
719:
714:
709:
703:
702:
698:
694:
690:
686:
682:
678:
674:
670:
666:
661:
660:
649:
648:
637:
636:
625:
624:
613:
612:
601:
600:
589:
588:
581:Special linear
577:
576:
569:General linear
565:
564:
559:
553:
544:
543:
540:
539:
536:
535:
530:
525:
517:
516:
503:
491:
478:
465:
463:Modular groups
461:
460:
459:
454:
441:
425:
422:
421:
416:
410:
409:
408:
405:
404:
399:
398:
397:
396:
391:
386:
383:
377:
376:
370:
369:
368:
367:
361:
360:
354:
353:
348:
339:
338:
336:Hall's theorem
333:
331:Sylow theorems
327:
326:
321:
313:
312:
311:
310:
304:
299:
296:Dihedral group
292:
291:
286:
280:
275:
269:
264:
253:
248:
247:
244:
243:
238:
237:
236:
235:
230:
222:
221:
220:
219:
214:
209:
204:
199:
194:
189:
187:multiplicative
184:
179:
174:
169:
161:
160:
159:
158:
153:
145:
144:
136:
135:
134:
133:
131:Wreath product
128:
123:
118:
116:direct product
110:
108:Quotient group
102:
101:
100:
99:
94:
89:
79:
76:
75:
72:
71:
63:
62:
15:
13:
10:
9:
6:
4:
3:
2:
8206:
8195:
8192:
8190:
8187:
8185:
8182:
8180:
8177:
8175:
8172:
8171:
8169:
8158:
8154:
8149:
8144:
8140:
8135:
8131:
8125:
8121:
8120:
8114:
8110:
8106:
8102:
8098:
8093:
8089:
8085:
8081:
8079:0-12-558180-7
8075:
8071:
8067:
8063:
8059:
8055:
8051:
8049:3-540-12179-X
8045:
8041:
8037:
8032:
8028:
8026:0-8176-4120-3
8022:
8017:
8016:
8009:
8008:
8004:
7996:
7992:
7988:
7986:3-7643-3092-9
7982:
7978:
7971:
7968:
7963:
7959:
7955:
7951:
7947:
7943:
7939:
7935:
7928:
7925:
7920:
7916:
7911:
7906:
7902:
7898:
7891:
7888:
7885:, Chapter 19.
7884:
7879:
7876:
7871:
7867:
7862:
7857:
7853:
7849:
7842:
7839:
7835:
7830:
7827:
7822:
7818:
7814:
7812:3-7643-6989-2
7808:
7804:
7797:
7794:
7789:
7785:
7781:
7777:
7773:
7769:
7764:
7759:
7755:
7751:
7744:
7741:
7737:
7732:
7729:
7725:
7724:Margulis 1991
7720:
7717:
7713:
7708:
7705:
7701:
7700:Margulis 1991
7696:
7693:
7688:
7684:
7680:
7673:
7670:
7665:
7661:
7657:
7653:
7649:
7645:
7641:
7637:
7630:
7623:
7620:
7616:
7611:
7608:
7604:
7599:
7596:
7592:
7587:
7584:
7580:
7575:
7572:
7568:
7563:
7560:
7555:
7551:
7547:
7543:
7539:
7535:
7530:
7525:
7521:
7517:
7510:
7507:
7501:
7499:
7485:
7465:
7459:
7439:
7419:
7413:
7393:
7373:
7350:
7316:
7304:
7302:
7280:
7254:
7246:
7244:
7242:
7226:
7206:
7186:
7166:
7140:
7129:
7115:
7112:
7108:
7092:
7080:
7078:
7064:
7048:
7031:
7008:
7006:
6987:
6965:
6949:
6926:
6904:
6900:
6884:
6864:
6852:
6848:Tree lattices
6847:
6845:
6842:
6840:
6811:
6789:
6787:
6758:
6733:
6731:
6729:
6725:
6720:
6719:
6718:Superrigidity
6714:
6712:
6708:
6707:George Mostow
6704:
6675:
6657:
6652:
6650:
6646:
6642:
6638:
6634:
6630:
6626:
6624:
6616:
6611:
6609:
6607:
6606:trace formula
6603:
6565:
6549:
6544:
6514:
6508:
6480:
6457:
6413:
6410:
6402:
6364:
6357:
6327:
6325:
6311:
6289:
6285:
6276:
6251:
6236:
6223:
6207:
6194:
6190:
6185:
6180:
6177:
6172:
6162:
6156:
6139:
6138:
6137:
6117:
6093:
6090:
6068:
6039:
6035:
6009:
6005:
5999:
5996:
5993:
5989:
5985:
5982:
5975:
5974:
5973:
5967:
5965:
5959:
5955:
5940:
5932:
5928:
5927:
5926:
5924:
5906:
5877:
5850:
5828:
5814:
5813:p-adic fields
5806:
5804:
5802:
5786:
5766:
5746:
5720:
5700:
5697:
5685:
5683:
5679:
5663:
5659:
5655:
5652:
5649:
5642:
5626:
5606:
5586:
5566:
5558:
5528:
5520:
5518:
5516:
5512:
5507:
5473:
5453:
5445:
5440:
5426:
5406:
5386:
5360:
5352:
5348:
5332:
5292:
5272:
5269:
5266:
5246:
5243:
5237:
5228:
5214:
5194:
5189:
5186:
5182:
5175:
5155:
5147:
5129:
5125:
5101:
5096:
5092:
5088:
5085:
5080:
5076:
5067:
5063:
5059:
5053:
5050:
5047:
5039:
5035:
5014:
5011:
5008:
4988:
4985:
4982:
4962:
4954:
4938:
4892:
4888:
4880:
4879:inner product
4864:
4852:
4847:
4845:
4843:
4839:
4835:
4827:
4807:
4804:
4801:
4787:
4781:
4778:
4775:
4756:
4755:
4754:
4751:
4744:
4741:
4740:
4739:
4712:
4709:
4706:
4692:
4686:
4683:
4680:
4661:
4657:
4654:
4650:
4647:
4646:open question
4631:
4628:
4625:
4622:
4616:
4613:
4610:
4576:
4573:
4570:
4556:
4550:
4547:
4544:
4510:
4507:
4504:
4485:
4484:
4483:
4462:
4458:
4435:
4432:
4427:
4423:
4415:
4411:
4396:
4393:
4390:
4367:
4364:
4361:
4350:
4331:
4328:
4325:
4306:
4291:
4288:
4285:
4262:
4259:
4256:
4246:of signature
4245:
4226:
4223:
4220:
4202:
4198:
4183:
4180:
4177:
4154:
4151:
4148:
4138:of signature
4137:
4118:
4115:
4112:
4094:
4090:
4089:
4088:
4086:
4082:
4066:
4063:
4055:
4039:
4031:
3993:
3985:
3977:
3975:
3959:
3952:
3948:
3944:
3939:
3935:
3912:
3905:
3901:
3897:
3892:
3888:
3867:
3841:
3835:
3832:
3829:
3820:
3787:
3774:
3758:
3720:
3702:
3664:
3660:
3655:
3651:
3648:
3645:
3638:
3634:
3629:
3600:
3581:
3577:
3572:
3568:
3565:
3562:
3555:
3551:
3546:
3517:
3516:
3515:
3501:
3498:
3475:
3453:
3449:
3445:
3442:
3439:
3434:
3430:
3426:
3423:
3414:
3412:
3396:
3393:
3388:
3380:
3375:
3348:
3344:
3340:
3335:
3327:
3322:
3318:
3314:
3309:
3284:
3262:
3258:
3254:
3249:
3245:
3241:
3238:
3218:
3206:
3204:
3180:
3175:
3148:
3135:
3119:
3089:
3061:
3057:
3028:
3015:
3012:
3009:
2964:
2956:
2905:
2887:
2871:
2861:
2853:
2848:
2846:
2844:
2839:
2832:
2830:
2827:
2810:
2804:
2801:
2793:
2789:
2773:
2764:
2717:
2712:
2709:
2705:
2684:
2644:
2621:
2615:
2612:
2604:
2564:
2520:
2488:
2472:
2428:
2419:
2416:
2400:
2388:
2383:
2381:
2379:
2374:
2372:
2371:affine groups
2368:
2364:
2359:
2357:
2348:
2346:
2344:
2343:commensurable
2339:
2312:
2299:
2283:
2240:
2197:
2180:
2179:modular group
2175:
2157:
2153:
2113:
2093:
2084:
2082:
2064:
2035:
2016:
2014:
2000:
1956:
1952:
1932:
1892:
1889:
1880:
1877:
1870:
1867:
1864:
1844:
1841:
1838:
1818:
1815:
1804:
1784:
1780:
1772:
1756:
1753:
1741:
1727:
1707:
1699:
1694:
1656:
1652:
1632:
1592:
1583:
1566:
1560:
1557:
1551:
1548:
1542:
1535:the equality
1518:
1514:
1511:
1508:
1488:
1485:
1482:
1462:
1439:
1436:
1426:
1422:
1416:
1392:
1388:
1368:
1361:
1360:Borel measure
1345:
1300:
1296:
1289:
1286:
1283:
1260:
1238:
1234:
1213:
1173:
1161:
1159:
1157:
1153:
1148:
1121:
1108:
1092:
1074:
1073:Radon measure
1070:
1066:
1047:
1037:
1032:
1016:
1014:
1010:
1006:
988:
959:
940:
935:
933:
931:
930:Cayley graphs
928:
924:
923:combinatorics
920:
916:
912:
908:
904:
900:
895:
893:
892:tree lattices
889:
885:
880:
878:
874:
870:
866:
861:
859:
855:
851:
847:
843:
839:
835:
831:
819:
814:
812:
807:
805:
800:
799:
797:
796:
789:
786:
785:
782:
779:
778:
775:
772:
771:
768:
765:
764:
761:
756:
755:
745:
742:
739:
738:
736:
730:
727:
725:
722:
721:
718:
715:
713:
710:
708:
705:
704:
701:
695:
693:
687:
685:
679:
677:
671:
669:
663:
662:
658:
654:
651:
650:
646:
642:
639:
638:
634:
630:
627:
626:
622:
618:
615:
614:
610:
606:
603:
602:
598:
594:
591:
590:
586:
582:
579:
578:
574:
570:
567:
566:
563:
560:
558:
555:
554:
551:
547:
542:
541:
534:
531:
529:
526:
524:
521:
520:
492:
467:
466:
464:
458:
455:
430:
427:
426:
420:
417:
415:
412:
411:
407:
406:
395:
392:
390:
387:
384:
381:
380:
379:
378:
375:
371:
366:
363:
362:
359:
356:
355:
352:
349:
347:
345:
341:
340:
337:
334:
332:
329:
328:
325:
322:
320:
317:
316:
315:
314:
308:
305:
302:
297:
294:
293:
289:
284:
281:
278:
273:
270:
267:
262:
259:
258:
257:
256:
251:
250:Finite groups
246:
245:
234:
231:
229:
226:
225:
224:
223:
218:
215:
213:
210:
208:
205:
203:
200:
198:
195:
193:
190:
188:
185:
183:
180:
178:
175:
173:
170:
168:
165:
164:
163:
162:
157:
154:
152:
149:
148:
147:
146:
143:
142:
137:
132:
129:
127:
124:
122:
119:
117:
114:
111:
109:
106:
105:
104:
103:
98:
95:
93:
90:
88:
85:
84:
83:
82:
77:Basic notions
74:
73:
69:
65:
64:
61:
56:
52:
47:
41:
36:
30:
26:
22:
8138:
8118:
8096:
8069:
8035:
8014:
7976:
7970:
7937:
7933:
7927:
7900:
7896:
7890:
7878:
7861:math/0111165
7851:
7847:
7841:
7829:
7802:
7796:
7763:math/0503088
7753:
7749:
7743:
7731:
7719:
7707:
7695:
7678:
7672:
7639:
7635:
7622:
7610:
7598:
7586:
7574:
7562:
7519:
7515:
7509:
7308:
7256:
7114:power series
7084:
7049:
7009:
7005:tree lattice
7004:
6856:
6843:
6790:
6737:
6727:
6723:
6716:
6715:
6711:Gopal Prasad
6702:
6653:
6648:
6644:
6627:
6622:
6620:
6545:
6356:number field
6331:
6274:
6272:
6026:
5971:
5963:
5957:
5930:
5922:
5810:
5686:
5557:Killing form
5530:
5515:nilmanifolds
5508:
5441:
5229:
4856:
4842:Property (T)
4831:
4757:Lattices in
4747:
4742:
4736:
4480:
3981:
3813:via the map
3684:
3415:
3410:
3210:
3178:
3176:
3062:states that
3056:Armand Borel
2863:
2840:
2836:
2828:
2765:
2420:
2392:
2375:
2360:
2352:
2340:
2176:
2174:is finite).
2085:
2020:
1802:
1770:
1742:
1698:Haar measure
1695:
1584:
1165:
1149:
1017:
944:
896:
891:
881:
869:local fields
862:
853:
833:
827:
656:
644:
632:
620:
608:
596:
584:
572:
418:
343:
300:
287:
276:
265:
261:Cyclic group
139:
126:Free product
97:Group action
60:Group theory
55:Group theory
54:
20:
5687:A subgroup
5680:are called
5146:tangent map
4307:The groups
3411:irreducible
2790:(and hence
1803:non-uniform
848:has finite
546:Topological
385:alternating
8194:Lie groups
8168:Categories
8005:References
7642:: 93–103.
6853:Definition
6401:adèle ring
6083:. Usually
4349:quaternion
4054:semisimple
2843:polycyclic
2826:elements.
2786:is always
2378:simplicity
2363:unimodular
2356:Lie groups
1945:such that
1769:is called
1743:A lattice
1273:such that
1162:Definition
830:Lie theory
653:Symplectic
593:Orthogonal
550:Lie groups
457:Free group
182:continuous
121:Direct sum
8148:1402.0962
7962:119638780
7910:1102.3574
7554:119130421
7529:1008.2911
7522:: 55–67.
7463:∖
7417:∖
7289:Γ
7265:Γ
7062:∖
7059:Γ
6903:biregular
6897:can be a
6572:∖
6515:⊂
6224:×
6195:⊂
6122:∞
6097:∞
5997:∈
5990:∏
5892:of split-
5801:orbifolds
5744:∖
5741:Γ
5698:⊂
5695:Γ
5494:Γ
5384:∖
5381:Γ
5313:Γ
5270:∈
5267:γ
5244:γ
5241:↦
5230:The maps
5187:−
5183:γ
5179:↦
5156:γ
5130:∗
5126:γ
5097:∗
5093:γ
5081:∗
5077:γ
5040:γ
5012:∈
5009:γ
4629:≥
4433:−
4394:≥
4289:≥
4181:≥
4064:±
3984:real rank
3945:−
3868:σ
3836:σ
3824:↦
3775:×
3652:×
3649:…
3646:×
3609:Γ
3598:is dense;
3569:×
3566:…
3563:×
3526:Γ
3499:⊂
3496:Γ
3446:×
3443:…
3440:×
3394:⊂
3385:Γ
3381:×
3372:Γ
3341:⊂
3332:Γ
3315:⊂
3306:Γ
3255:×
3136:⊂
3070:Γ
3016:∩
3007:Γ
2805:
2724:Γ
2718:
2710:−
2665:Γ
2616:
2300:⊂
2162:Γ
2134:Γ
1981:Γ
1961:Γ
1913:Γ
1893:γ
1884:Γ
1881:∈
1878:γ
1871:⋃
1842:⊂
1816:⊂
1813:Γ
1789:Γ
1754:⊂
1751:Γ
1681:Γ
1661:Γ
1613:Γ
1561:μ
1543:μ
1523:Γ
1512:⊂
1486:∈
1443:∞
1431:Γ
1417:μ
1397:Γ
1369:μ
1326:Γ
1284:∩
1281:Γ
1194:Γ
1109:⊂
1038:⊂
1009:countable
927:expanding
717:Conformal
605:Euclidean
212:nilpotent
7788:18414386
7664:55721623
6623:rigidity
6612:Rigidity
5925:. Then:
1318:). Then
712:Poincaré
557:Solenoid
429:Integers
419:Lattices
394:sporadic
389:Lie type
217:solvable
207:dihedral
192:additive
177:infinite
87:Subgroup
8153:Bibcode
8109:0507234
8088:1278263
8058:1090825
7995:0670072
7954:1132296
7821:1978431
7780:2067695
7687:1241644
7656:0932135
7546:2881324
7111:Laurent
6899:regular
5676:: such
4081:isogeny
4028:-split
2605:) then
2369:or the
2223:inside
1771:uniform
1065:compact
834:lattice
707:Lorentz
629:Unitary
528:Lattice
468:PSL(2,
202:abelian
113:(Semi-)
8126:
8107:
8086:
8076:
8056:
8046:
8023:
7993:
7983:
7960:
7952:
7819:
7809:
7786:
7778:
7685:
7662:
7654:
7552:
7544:
7159:) and
6645:v>0
6027:where
5921:-rank
5347:freely
5117:where
4748:Using
4383:) for
3860:where
1455:) and
913:), in
905:), in
562:Circle
493:SL(2,
382:cyclic
346:-group
197:cyclic
172:finite
167:simple
151:kernel
23:, see
8143:arXiv
7958:S2CID
7905:arXiv
7856:arXiv
7784:S2CID
7758:arXiv
7660:S2CID
7632:(PDF)
7550:S2CID
7524:arXiv
7502:Notes
5559:. If
4526:, in
4030:torus
2697:then
1857:with
1067:, or
919:flows
888:trees
867:over
840:is a
836:in a
746:Sp(∞)
743:SU(∞)
156:image
8124:ISBN
8074:ISBN
8044:ISBN
8021:ISBN
7981:ISBN
7901:2011
7807:ISBN
6857:Let
6709:and
6654:The
6546:The
6399:its
6377:and
5513:and
5442:The
5349:and
5259:for
5148:(at
5027:put
4412:The
4278:for
4199:The
4170:for
4091:The
3982:The
3058:and
1437:<
1166:Let
1007:and
740:O(∞)
729:Loop
548:and
7942:doi
7915:doi
7866:doi
7852:124
7768:doi
7754:114
7644:doi
7534:doi
7257:If
7199:of
7085:If
6942:of
6901:or
6473:of
6332:If
5958:r=1
5956:if
5929:If
5207:of
4955:on
4857:If
4242:of
4134:of
4087:):
4032:of
3927:to
2953:of
2888:in
2864:If
2802:dim
2763:.
2706:exp
2613:exp
2489:of
2465:of
2086:If
1253:of
894:).
828:In
655:Sp(
643:SU(
619:SO(
583:SL(
571:GL(
8170::
8151:.
8105:MR
8103:.
8084:MR
8082:.
8054:MR
8052:.
7991:MR
7989:.
7956:.
7950:MR
7948:.
7936:.
7913:.
7899:.
7864:.
7850:.
7817:MR
7815:.
7782:.
7776:MR
7774:.
7766:.
7752:.
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7658:.
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7650:.
7640:66
7638:.
7634:.
7548:.
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6841:.
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5227:.
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4436:20
3974:.
3174:.
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2013:.
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879:.
631:U(
607:E(
595:O(
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8159:.
8155::
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8132:.
8111:.
8090:.
8060:.
8029:.
7997:.
7964:.
7944::
7938:1
7921:.
7917::
7907::
7872:.
7868::
7858::
7823:.
7790:.
7770::
7760::
7689:.
7666:.
7646::
7556:.
7536::
7526::
7486:H
7466:T
7460:H
7440:H
7420:T
7414:H
7394:H
7374:H
7354:)
7351:T
7348:(
7344:t
7341:u
7338:A
7317:H
7227:G
7207:F
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7167:G
7147:)
7144:)
7141:t
7138:(
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7130:p
7125:F
7093:F
7065:T
7035:)
7032:T
7029:(
7025:t
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6988:T
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6924:(
6920:t
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6825:)
6821:C
6817:(
6812:2
6807:L
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6801:P
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6768:R
6764:(
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6724:G
6689:)
6685:R
6681:(
6676:2
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6584:A
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6094:=
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4802:n
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4502:(
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2001:G
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