3911:
3007:)) by eigenvalues and nilpotence (concretely, nilpotence means where 1s occur in the Jordan blocks). Thus elements of SL(2) are classified up to conjugacy in GL(2) (or indeed SL(2)) by trace (since determinant is fixed, and trace and determinant determine eigenvalues), except if the eigenvalues are equal, so ±I and the parabolic elements of trace +2 and trace -2 are not conjugate (the former have no off-diagonal entries in Jordan form, while the latter do).
52:
3244:
4125:
1926:
3011:
conjugacy classes for each trace (clockwise and counterclockwise rotations), for absolute value of the trace equal to 2 there are three conjugacy classes for each trace (positive shear, identity, negative shear), and for absolute value of the trace greater than 2 there is one conjugacy class for a given trace.
1000:
3010:
Up to conjugacy in SL(2) (instead of GL(2)), there is an additional datum, corresponding to orientation: a clockwise and counterclockwise (elliptical) rotation are not conjugate, nor are a positive and negative shear, as detailed above; thus for absolute value of trace less than 2, there are two
3498:
3375:
3082:
2976:
2741:
1658:
3983:
1751:
2155:
2855:
2795:
1508:
852:
2060:
2986:
of half of the trace, but the sign can be positive or negative: in contrast to the elliptic case, a squeeze and its inverse are conjugate in SL₂ (by a rotation in the axes; for standard axes, a rotation by 90°).
3381:
3893:
3839:
3785:
3728:
3667:
3613:
3250:
3239:{\displaystyle \mathbf {K} =\left\{{\begin{pmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{pmatrix}}\in SL(2,\mathbb {R} )\ |\ \theta \in \mathbf {R} \right\}\cong SO(2),}
2531:
these sets are not closed under multiplication (the product of two parabolic elements need not be parabolic, and so forth). However, each element is conjugate to a member of one of 3 standard
4653:
3074:
2342:
2212:
2910:
2277:
2680:
1558:
2464:
2412:
4120:{\displaystyle {\overline {\mathrm {SL} (2,\mathbf {R} )}}\to \cdots \to \mathrm {Mp} (2,\mathbf {R} )\to \mathrm {SL} (2,\mathbf {R} )\to \mathrm {PSL} (2,\mathbf {R} ).}
497:
472:
435:
2438:
1921:{\displaystyle {\begin{bmatrix}a&b\\c&d\end{bmatrix}}\mapsto {\begin{bmatrix}a^{2}&2ab&b^{2}\\ac&ad+bc&bd\\c^{2}&2cd&d^{2}\end{bmatrix}}.}
2071:
3955:
of the modular group. These are lattices inside the relevant algebraic groups, and this corresponds algebraically to the universal covering group in topology.
2800:
2384:|tr|; dividing by 2 corrects for the effect of dimension, while absolute value corresponds to ignoring an overall factor of ±1 such as when working in PSL(2,
4348:) has no nontrivial finite-dimensional unitary representations. This is a feature of every connected simple non-compact Lie group. For outline of proof, see
995:{\displaystyle {\mbox{SL}}(2,\mathbf {R} )=\left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\colon a,b,c,d\in \mathbf {R} {\mbox{ and }}ad-bc=1\right\}.}
2746:
2608:
Elements of trace 0 may be called "circular elements" (by analogy with eccentricity) but this is rarely done; they correspond to elements with eigenvalues ±
1318:
4349:
2502:
A subgroup that is contained with the elliptic (respectively, parabolic, hyperbolic) elements, plus the identity and negative identity, is called an
2648:
2631:
of half of the trace, with the sign of the rotation determined by orientation. (A rotation and its inverse are conjugate in GL(2) but not SL(2).)
2570:
of the
Euclidean plane – they can be interpreted as rotations in a possibly non-orthogonal basis – and the corresponding element of PSL(2,
799:
1996:
2473:) they are the same), have trace ±2, and hence by this classification are parabolic elements, though they are often considered separately.
3493:{\displaystyle \mathbf {N} =\left\{{\begin{pmatrix}1&x\\0&1\end{pmatrix}}\in SL(2,\mathbb {R} )\ |\ x\in \mathbf {R} \right\}.}
3788:
1282:) (as linear transforms of the torus), and these interpretations can also be viewed in light of the general theory of SL(2,
3370:{\displaystyle \mathbf {A} =\left\{{\begin{pmatrix}r&0\\0&r^{-1}\end{pmatrix}}\in SL(2,\mathbb {R} )\ |\ r>0\right\},}
2496:
3848:
3794:
3740:
3683:
3622:
3568:
357:
4467:
307:
1725:(remember that the Lie algebra elements are also 2 × 2 matrices), yielding a faithful 3-dimensional linear
792:
302:
4359:) is quite interesting. The group has several families of unitary representations, which were worked out in detail by
1135:), the group of real 2 × 2 matrices with determinant ±1; this is more commonly used in the context of the
4696:
4669:
4615:
2495:) (real Möbius transformations), with the addition of "loxodromic" transformations corresponding to complex traces;
1686:
of hyperbolic space, and the corresponding Möbius transformations of the disc are the hyperbolic isometries of the
1047:
4281:
3038:
2476:
1062:
4337:
2971:{\displaystyle \left({\begin{smallmatrix}\lambda \\&\lambda ^{-1}\end{smallmatrix}}\right)\times \{\pm I\}}
718:
2542:, as do the hyperbolic elements (excluding ±1). By contrast, the parabolic elements, together with ±1, form a
2297:
2167:
785:
4529:
Gelfand, Israel
Moiseevich; Neumark, Mark Aronovich (1946). "Unitary representations of the Lorentz group".
4738:
4733:
3503:
These three elements are the generators of the
Elliptic, Hyperbolic, and Parabolic subsets respectively.
2232:
2160:
This leads to the following classification of elements, with corresponding action on the
Euclidean plane:
1683:
1529:
4268:
is much larger than the universal covering group. However these large central extensions do not take the
3952:
3925:
2736:{\displaystyle \left({\begin{smallmatrix}1&\lambda \\&1\end{smallmatrix}}\right)\times \{\pm I\}}
1259:
402:
216:
3562:
1653:{\displaystyle z\mapsto {\frac {az+b}{cz+d}}\;\;\;\;{\mbox{ (where }}a,b,c,d\in \mathbf {R} {\mbox{)}}.}
4158:
2639:
A parabolic element has only a single eigenvalue, which is either 1 or -1. Such an element acts as a
2365:
1722:
134:
1179:
4414:
1988:
1236:
1206:
1202:
1160:
2443:
2391:
4265:
4238:
2624:
2617:
1043:
600:
334:
211:
99:
2488:
1522:
480:
455:
418:
4395:
3670:
2417:
1672:
17:
4326:
4322:
2882:
2878:
2652:
2532:
2223:
750:
540:
1687:
4723:
4492:
4181:
3020:
2897:
2628:
1726:
1152:
1028:
624:
4743:
4706:
4679:
4585:
4563:
4542:
4521:
4477:
4390:
3674:
2623:
Elliptic elements are conjugate into the subgroup of rotations of the
Euclidean plane, the
2598:
2150:{\displaystyle \lambda ={\frac {\mathrm {tr} (A)\pm {\sqrt {\mathrm {tr} (A)^{2}-4}}}{2}}.}
1698:
1303:
1210:
839:
817:
564:
552:
170:
104:
4691:. Princeton Mathematical Series. Vol. 35. Princeton, NJ: Princeton University Press.
2538:
Topologically, as trace is a continuous map, the elliptic elements (excluding ±1) form an
8:
4728:
4452:
Geometry of Möbius transformations. Elliptic, parabolic and hyperbolic actions of SL(2,R)
4424:
4245:), and the circle SO(2) / {±1} is a maximal compact subgroup of PSL(2,
3904:
3532:
3520:
1702:
832:
139:
34:
4567:
4490:
Bargmann, Valentine (1947). "Irreducible
Unitary Representations of the Lorentz Group".
2869:
for a hyperbolic element are both real, and are reciprocals. Such an element acts as a
2850:{\displaystyle \left({\begin{smallmatrix}-1&\pm 1\\&-1\end{smallmatrix}}\right)}
4594:
4551:
4509:
2996:
124:
96:
2790:{\displaystyle \left({\begin{smallmatrix}1&\pm 1\\&1\end{smallmatrix}}\right)}
1503:{\displaystyle \mapsto {\begin{pmatrix}a&c\\b&d\end{pmatrix}}\ =\ \ =\,\left.}
4692:
4665:
4599:
4463:
4269:
3963:
3731:
3528:
3512:
2903:
Hyperbolic elements are conjugate into the 2 component group of standard squeezes × ±
2559:
1964:
1537:
1167:
1117:
529:
372:
266:
695:
4657:
4589:
4571:
4501:
4455:
4302:
4253:
4234:
2979:
1940:
1664:
1244:
1183:
1171:
1125:
1012:
680:
672:
664:
656:
648:
636:
576:
516:
506:
348:
290:
165:
3023:
of a group is a method to construct the group as a product of three Lie subgroups
2673:
Parabolic elements are conjugate into the 2 component group of standard shears × ±
1675:, it is also isomorphic to the group of conformal automorphisms of the unit disc.
4702:
4675:
4581:
4538:
4517:
4473:
4212:
3842:
2886:
2870:
2656:
2579:
2353:
1956:
1738:
1255:
1094:
1006:
764:
757:
743:
700:
588:
511:
341:
255:
195:
75:
2920:
2810:
2756:
2690:
4419:
4372:
4360:
4208:
4189:
4166:
3730:
is a line bundle over the hyperbolic plane. When imbued with a left-invariant
2508:
1518:
1109:
1055:
771:
707:
397:
377:
314:
279:
200:
190:
175:
160:
114:
91:
4661:
4717:
4400:
4220:
3929:
3896:
3524:
2893:
2663:
2640:
2594:
2586:
2523:) into elliptic, parabolic, and hyperbolic elements is a classification into
2361:
2288:
1948:
1251:
1222:
1136:
1098:
1009:
690:
612:
446:
319:
185:
4227:), which acts on a tessellation of the hyperbolic plane by ideal triangles.
3910:
4656:. Graduate Texts in Mathematics. Vol. 105. New York: Springer-Verlag.
4603:
4385:
4364:
4231:
4216:
4197:
3616:
3555:
2368:: if one defines eccentricity as half the absolute value of the trace (ε =
1932:
1671:) is the group of conformal automorphisms of the upper half-plane. By the
1278:) have additional interpretations, as do elements of the group SL(2,
1225:
1051:
545:
244:
233:
180:
155:
150:
109:
80:
43:
4576:
4429:
4368:
4310:
3941:
3915:
2866:
2743:. In fact, they are all conjugate (in SL(2)) to one of the four matrices
2563:
2555:
2469:
The identity element 1 and negative identity element −1 (in PSL(2,
1694:
843:
836:
813:
4513:
3735:
2667:
2543:
2055:{\displaystyle \lambda ^{2}\,-\,\mathrm {tr} (A)\,\lambda \,+\,1\,=\,0}
1976:
1693:
The above formula can be also used to define Möbius transformations of
1679:
1299:
1267:
712:
440:
2873:
of the
Euclidean plane, and the corresponding element of PSL(2,
2643:
on the
Euclidean plane, and the corresponding element of PSL(2,
1733:). This can alternatively be described as the action of PSL(2,
1663:
This is precisely the set of Möbius transformations that preserve the
1229:
1197:) has several interesting descriptions, up to Lie group isomorphism:
1016:
533:
4505:
3936:
Under this covering, the preimage of the modular group PSL(2,
3900:
2857:(in GL(2) or SL(2), the ± can be omitted, but in SL(2) it cannot).
2575:
2567:
2539:
1240:
1024:
1020:
70:
4305:, and is the split-real form of the complex Lie group SL(2,
1513:
These projective transformations form a subgroup of PSL(2,
4459:
4405:
1032:
412:
326:
3615:, is an example of a finite-dimensional Lie group that is not a
3544:
2983:
2597:. These are all the elements of the modular group with finite
1701:. The corresponding geometries are in non-trivial relations to
51:
4355:
The infinite-dimensional representation theory of SL(2,
2602:
4332:
The finite-dimensional representation theory of SL(2,
3958:
The 2-fold covering group can be identified as Mp(2,
1163:
4130:
However, there are other covering groups of PSL(2,
3888:{\displaystyle {\overline {{\mbox{SL}}(2,\mathbf {R} )}}}
3841:
is the universal cover of the unit tangent bundle to any
3834:{\displaystyle {\overline {{\mbox{SL}}(2,\mathbf {R} )}}}
3780:{\displaystyle {\overline {{\mbox{SL}}(2,\mathbf {R} )}}}
3723:{\displaystyle {\overline {{\mbox{SL}}(2,\mathbf {R} )}}}
3662:{\displaystyle {\overline {{\mbox{SL}}(2,\mathbf {R} )}}}
3608:{\displaystyle {\overline {{\mbox{SL}}(2,\mathbf {R} )}}}
1190:) preserves unoriented area: it may reverse orientation.
2612:, and are conjugate to rotation by 90°, and square to -
1943:(2,1), and induces an isomorphism between PSL(2,
4552:"Plancherel formula for the 2×2 real unimodular group"
4211:. These are the hyperbolic analogue of the Euclidean
3856:
3802:
3748:
3691:
3630:
3576:
3403:
3272:
3104:
3043:
1799:
1760:
1641:
1605:
1552:) act on the complex plane by Möbius transformations:
1363:
957:
893:
857:
4618:
3986:
3851:
3797:
3743:
3686:
3625:
3571:
3384:
3253:
3085:
3041:
2913:
2803:
2749:
2683:
2446:
2420:
2394:
2300:
2235:
2170:
2074:
1999:
1754:
1561:
1528:
When the real line is considered the boundary of the
1321:
1182:. It is also isomorphic to the group of unit-length
855:
483:
458:
421:
3977:
The aforementioned groups together form a sequence:
4454:. London: Imperial College Press. p. xiv+192.
4647:
4119:
3887:
3833:
3779:
3722:
3661:
3607:
3492:
3369:
3238:
3068:
2970:
2849:
2789:
2735:
2558:for an elliptic element are both complex, and are
2458:
2432:
2406:
2336:
2271:
2206:
2149:
2054:
1959:restricts to the isometric action of PSL(2,
1920:
1652:
1502:
994:
491:
466:
429:
2999:, matrices are classified up to conjugacy (in GL(
4715:
4340:, which is the compact real form of SL(2,
27:Group of real 2×2 matrices with unit determinant
4689:Three-dimensional geometry and topology. Vol. 1
4648:{\displaystyle \mathrm {SL} _{2}(\mathbf {R} )}
3506:
3014:
1745:. The result is the following representation:
4549:
4528:
2360:The names correspond to the classification of
4687:Thurston, William (1997). Silvio Levy (ed.).
1970:
793:
4272:into account and are somewhat pathological.
2965:
2956:
2730:
2721:
3069:{\displaystyle {\mbox{SL}}(2,\mathbf {R} )}
1274:Elements of the modular group PSL(2,
3543:), and can be thought of as the bundle of
1603:
1602:
1601:
1600:
800:
786:
4593:
4575:
4188:) is the two-element group {±1}, and the
4161:by divisibility; these cover SL(2,
3453:
3332:
3181:
2048:
2044:
2040:
2036:
2032:
2014:
2010:
1708:
1543:
1445:
1258:. Equivalently, it is isomorphic to the
485:
460:
423:
4686:
4489:
4275:
3909:
2535:(possibly times ±1), as detailed below.
1678:These Möbius transformations act as the
4325:2 × 2 matrices. It is the
2982:of the hyperbolic rotation is given by
2337:{\displaystyle |\mathrm {tr} (A)|>2}
2207:{\displaystyle |\mathrm {tr} (A)|<2}
14:
4716:
4175:
3970:) as the symplectic group Sp(2,
3535:on the hyperbolic plane. SL(2,
2860:
1951:SO(2,1). This action of PSL(2,
1717:) acts on its Lie algebra sl(2,
1262:SO(1,2). It follows that SL(2,
358:Classification of finite simple groups
4449:
3550:The fundamental group of SL(2,
2634:
2566:. Such an element is conjugate to a
1131:Another related group is SL(2,
4610:
3539:) is a 2-fold cover of PSL(2,
2990:
2589:must have eigenvalues {ω, ω}, where
2549:
2272:{\displaystyle |\mathrm {tr} (A)|=2}
1019:of dimension 3 with applications in
4256:of the discrete group PSL(2,
4219:. The most famous of these is the
4203:Discrete subgroups of PSL(2,
3899:over some 2-dimensional hyperbolic
1108:Also closely related is the 2-fold
24:
4624:
4621:
4443:
4093:
4090:
4087:
4062:
4059:
4034:
4031:
3994:
3991:
3523:of the hyperbolic plane. It is a
2310:
2307:
2245:
2242:
2180:
2177:
2110:
2107:
2088:
2085:
2019:
2016:
1699:double (aka split-complex) numbers
25:
4755:
2919:
2881:of the hyperbolic plane and as a
2809:
2755:
2689:
2651:of the hyperbolic plane and as a
1048:fractional linear transformations
4638:
4350:non-unitarity of representations
4107:
4076:
4048:
4008:
3872:
3818:
3764:
3707:
3646:
3592:
3478:
3386:
3255:
3206:
3087:
3059:
2627:SO(2); the angle of rotation is
2593:is a primitive 3rd, 4th, or 6th
1667:. It follows that PSL(2,
1636:
1178:) and the special unitary group
952:
873:
50:
4321:), is the algebra of all real,
2578:of the hyperbolic plane and of
1289:
1142:
1063:projective special linear group
4642:
4634:
4344:). In particular, SL(2,
4338:representation theory of SU(2)
4111:
4097:
4083:
4080:
4066:
4055:
4052:
4038:
4027:
4021:
4012:
3998:
3876:
3862:
3822:
3808:
3768:
3754:
3711:
3697:
3650:
3636:
3596:
3582:
3464:
3457:
3443:
3343:
3336:
3322:
3230:
3224:
3192:
3185:
3171:
3063:
3049:
2519:The trichotomy of SL(2,
2459:{\displaystyle \epsilon >1}
2407:{\displaystyle \epsilon <1}
2324:
2320:
2314:
2302:
2259:
2255:
2249:
2237:
2194:
2190:
2184:
2172:
2121:
2114:
2098:
2092:
2029:
2023:
1791:
1565:
1436:
1403:
1355:
1340:
1337:
1334:
1322:
1093:denotes the 2 × 2
877:
863:
719:Infinite dimensional Lie group
13:
1:
4436:
2605:as periodic diffeomorphisms.
4556:Proc. Natl. Acad. Sci. U.S.A
4016:
3880:
3826:
3772:
3715:
3654:
3600:
3507:Topology and universal cover
3015:Iwasawa or KAN decomposition
2616:: they are the non-identity
492:{\displaystyle \mathbb {Z} }
467:{\displaystyle \mathbb {Z} }
430:{\displaystyle \mathbb {Z} }
7:
4450:Kisil, Vladimir V. (2012).
4378:
3953:universal central extension
3926:universal central extension
3845:. Any manifold modeled on
2892:Hyperbolic elements of the
2574:) acts as (conjugate to) a
2433:{\displaystyle \epsilon =1}
1260:indefinite orthogonal group
217:List of group theory topics
10:
4760:
4410:) (Möbius transformations)
4279:
4159:lattice of covering groups
3966:, thinking of SL(2,
3519:) can be described as the
3076:these three subgroups are
2662:Parabolic elements of the
1971:Classification of elements
1207:projective transformations
1193:The quotient PSL(2,
1077:) = SL(2,
4662:10.1007/978-1-4612-5142-2
4415:Projective transformation
4301:) is a real, non-compact
4282:Representation theory of
3789:eight Thurston geometries
3547:on the hyperbolic plane.
2585:Elliptic elements of the
2497:analogous classifications
1989:characteristic polynomial
1967:of the hyperbolic plane.
1548:Elements of PSL(2,
1294:Elements of PSL(2,
1120:(thinking of SL(2,
4531:Acad. Sci. USSR. J. Phys
4239:maximal compact subgroup
3895:is orientable, and is a
3680:As a topological space,
3563:universal covering group
2625:special orthogonal group
2479:is used for SL(2,
1186:. The group SL(2,
1044:complex upper half-plane
335:Elementary abelian group
212:Glossary of group theory
4550:Harish-Chandra (1952).
4396:Projective linear group
4336:) is equivalent to the
4317:), denoted sl(2,
4134:) corresponding to all
2533:one-parameter subgroups
2477:The same classification
1673:Riemann mapping theorem
1266:) is isomorphic to the
1061:(the 2 × 2
4649:
4260:) is much larger than
4121:
3933:
3889:
3835:
3781:
3724:
3663:
3609:
3494:
3371:
3240:
3070:
2972:
2898:Anosov diffeomorphisms
2851:
2791:
2737:
2601:, and they act on the
2489:Möbius transformations
2460:
2434:
2408:
2388:)), then this yields:
2338:
2273:
2222:and is conjugate to a
2208:
2151:
2056:
1922:
1713:The group SL(2,
1709:Adjoint representation
1703:Lobachevskian geometry
1684:upper half-plane model
1654:
1544:Möbius transformations
1523:Möbius transformations
1504:
1153:linear transformations
1151:) is the group of all
1069:). More specifically,
996:
751:Linear algebraic group
493:
468:
431:
4650:
4577:10.1073/pnas.38.4.337
4493:Annals of Mathematics
4276:Representation theory
4122:
3913:
3890:
3836:
3782:
3725:
3673:, finite-dimensional
3664:
3610:
3495:
3372:
3241:
3071:
3021:Iwasawa decomposition
2973:
2852:
2792:
2738:
2461:
2435:
2409:
2339:
2274:
2209:
2152:
2057:
1923:
1655:
1517:), which acts on the
1505:
1254:of three-dimensional
1250:It is the restricted
1029:representation theory
997:
835:of 2 × 2
494:
469:
432:
4616:
4611:Lang, Serge (1985).
4391:Special linear group
4264:, and the universal
3984:
3849:
3795:
3741:
3684:
3623:
3569:
3533:symplectic structure
3527:, and has a natural
3382:
3251:
3083:
3039:
2911:
2801:
2747:
2681:
2499:are used elsewhere.
2444:
2418:
2392:
2298:
2233:
2168:
2072:
1997:
1752:
1559:
1319:
1304:real projective line
1211:real projective line
1054:factors through the
853:
818:special linear group
481:
456:
419:
4739:Hyperbolic geometry
4734:Projective geometry
4568:1952PNAS...38..337H
4425:Table of Lie groups
4176:Algebraic structure
3905:Seifert fiber space
3787:becomes one of the
3521:unit tangent bundle
2861:Hyperbolic elements
2514:hyperbolic subgroup
2491:) and PSL(2,
2483:) and PSL(2,
1688:Poincaré disk model
1235:It is the group of
1221:It is the group of
1201:It is the group of
125:Group homomorphisms
35:Algebraic structure
4645:
4117:
3934:
3885:
3860:
3843:hyperbolic surface
3831:
3806:
3777:
3752:
3720:
3695:
3659:
3634:
3605:
3580:
3554:) is the infinite
3490:
3428:
3367:
3307:
3236:
3156:
3066:
3047:
2997:Jordan normal form
2968:
2947:
2946:
2847:
2841:
2840:
2787:
2781:
2780:
2733:
2712:
2711:
2635:Parabolic elements
2546:that is not open.
2509:parabolic subgroup
2456:
2430:
2404:
2334:
2269:
2204:
2147:
2052:
1918:
1909:
1785:
1737:) on the space of
1650:
1645:
1609:
1607: (where
1538:hyperbolic motions
1500:
1388:
1097:. It contains the
1081:) / {±
992:
961:
918:
861:
601:Special orthogonal
489:
464:
427:
308:Lagrange's theorem
4469:978-1-84816-858-9
4266:central extension
4157:)), which form a
4019:
3964:metaplectic group
3944:on 3 generators,
3883:
3859:
3829:
3805:
3775:
3751:
3718:
3694:
3657:
3633:
3603:
3579:
3529:contact structure
3513:topological space
3470:
3462:
3349:
3341:
3198:
3190:
3046:
2991:Conjugacy classes
2550:Elliptic elements
2504:elliptic subgroup
2142:
2136:
1965:hyperboloid model
1644:
1608:
1598:
1488:
1481:
1441:
1423:
1402:
1396:
1351:
1118:metaplectic group
960:
860:
810:
809:
385:
384:
267:Alternating group
224:
223:
16:(Redirected from
4751:
4710:
4683:
4654:
4652:
4651:
4646:
4641:
4633:
4632:
4627:
4607:
4597:
4579:
4546:
4525:
4482:
4481:
4447:
4303:simple Lie group
4254:Schur multiplier
4213:wallpaper groups
4126:
4124:
4123:
4118:
4110:
4096:
4079:
4065:
4051:
4037:
4020:
4015:
4011:
3997:
3988:
3894:
3892:
3891:
3886:
3884:
3879:
3875:
3861:
3857:
3853:
3840:
3838:
3837:
3832:
3830:
3825:
3821:
3807:
3803:
3799:
3791:. For example,
3786:
3784:
3783:
3778:
3776:
3771:
3767:
3753:
3749:
3745:
3729:
3727:
3726:
3721:
3719:
3714:
3710:
3696:
3692:
3688:
3668:
3666:
3665:
3660:
3658:
3653:
3649:
3635:
3631:
3627:
3614:
3612:
3611:
3606:
3604:
3599:
3595:
3581:
3577:
3573:
3499:
3497:
3496:
3491:
3486:
3482:
3481:
3468:
3467:
3460:
3456:
3433:
3432:
3389:
3376:
3374:
3373:
3368:
3363:
3359:
3347:
3346:
3339:
3335:
3312:
3311:
3304:
3303:
3258:
3245:
3243:
3242:
3237:
3214:
3210:
3209:
3196:
3195:
3188:
3184:
3161:
3160:
3090:
3075:
3073:
3072:
3067:
3062:
3048:
3044:
2980:hyperbolic angle
2977:
2975:
2974:
2969:
2952:
2948:
2943:
2942:
2929:
2856:
2854:
2853:
2848:
2846:
2842:
2830:
2796:
2794:
2793:
2788:
2786:
2782:
2773:
2742:
2740:
2739:
2734:
2717:
2713:
2704:
2465:
2463:
2462:
2457:
2439:
2437:
2436:
2431:
2413:
2411:
2410:
2405:
2383:
2381:
2380:
2377:
2374:
2343:
2341:
2340:
2335:
2327:
2313:
2305:
2278:
2276:
2275:
2270:
2262:
2248:
2240:
2213:
2211:
2210:
2205:
2197:
2183:
2175:
2156:
2154:
2153:
2148:
2143:
2138:
2137:
2129:
2128:
2113:
2105:
2091:
2082:
2061:
2059:
2058:
2053:
2022:
2009:
2008:
1927:
1925:
1924:
1919:
1914:
1913:
1906:
1905:
1883:
1882:
1834:
1833:
1811:
1810:
1790:
1789:
1729:of PSL(2,
1665:upper half-plane
1659:
1657:
1656:
1651:
1646:
1642:
1639:
1610:
1606:
1599:
1597:
1583:
1569:
1530:hyperbolic plane
1509:
1507:
1506:
1501:
1496:
1492:
1486:
1482:
1480:
1466:
1452:
1439:
1421:
1400:
1394:
1393:
1392:
1349:
1311:
1245:hyperbolic plane
1218:
1172:symplectic group
1126:symplectic group
1001:
999:
998:
993:
988:
984:
962:
958:
955:
923:
922:
876:
862:
858:
802:
795:
788:
744:Algebraic groups
517:Hyperbolic group
507:Arithmetic group
498:
496:
495:
490:
488:
473:
471:
470:
465:
463:
436:
434:
433:
428:
426:
349:Schur multiplier
303:Cauchy's theorem
291:Quaternion group
239:
238:
65:
64:
54:
41:
30:
29:
21:
4759:
4758:
4754:
4753:
4752:
4750:
4749:
4748:
4714:
4713:
4699:
4672:
4637:
4628:
4620:
4619:
4617:
4614:
4613:
4506:10.2307/1969129
4486:
4485:
4470:
4448:
4444:
4439:
4434:
4381:
4327:Bianchi algebra
4313:of SL(2,
4295:
4287:
4278:
4241:of SL(2,
4209:Fuchsian groups
4184:of SL(2,
4178:
4152:
4106:
4086:
4075:
4058:
4047:
4030:
4007:
3990:
3989:
3987:
3985:
3982:
3981:
3951:, which is the
3950:
3923:
3871:
3855:
3854:
3852:
3850:
3847:
3846:
3817:
3801:
3800:
3798:
3796:
3793:
3792:
3763:
3747:
3746:
3744:
3742:
3739:
3738:
3706:
3690:
3689:
3687:
3685:
3682:
3681:
3645:
3629:
3628:
3626:
3624:
3621:
3620:
3591:
3575:
3574:
3572:
3570:
3567:
3566:
3531:induced by the
3515:, PSL(2,
3509:
3477:
3463:
3452:
3427:
3426:
3421:
3415:
3414:
3409:
3399:
3398:
3397:
3393:
3385:
3383:
3380:
3379:
3342:
3331:
3306:
3305:
3296:
3292:
3290:
3284:
3283:
3278:
3268:
3267:
3266:
3262:
3254:
3252:
3249:
3248:
3205:
3191:
3180:
3155:
3154:
3143:
3131:
3130:
3116:
3100:
3099:
3098:
3094:
3086:
3084:
3081:
3080:
3058:
3042:
3040:
3037:
3036:
3017:
2993:
2945:
2944:
2935:
2931:
2927:
2926:
2918:
2914:
2912:
2909:
2908:
2887:Minkowski space
2871:squeeze mapping
2863:
2839:
2838:
2828:
2827:
2819:
2808:
2804:
2802:
2799:
2798:
2779:
2778:
2771:
2770:
2762:
2754:
2750:
2748:
2745:
2744:
2710:
2709:
2702:
2701:
2696:
2688:
2684:
2682:
2679:
2678:
2657:Minkowski space
2637:
2580:Minkowski space
2552:
2506:(respectively,
2445:
2442:
2441:
2419:
2416:
2415:
2393:
2390:
2389:
2378:
2375:
2372:
2371:
2369:
2354:squeeze mapping
2323:
2306:
2301:
2299:
2296:
2295:
2258:
2241:
2236:
2234:
2231:
2230:
2193:
2176:
2171:
2169:
2166:
2165:
2124:
2120:
2106:
2104:
2084:
2083:
2081:
2073:
2070:
2069:
2015:
2004:
2000:
1998:
1995:
1994:
1973:
1957:Minkowski space
1935:on sl(2,
1908:
1907:
1901:
1897:
1895:
1884:
1878:
1874:
1871:
1870:
1862:
1845:
1836:
1835:
1829:
1825:
1823:
1812:
1806:
1802:
1795:
1794:
1784:
1783:
1778:
1772:
1771:
1766:
1756:
1755:
1753:
1750:
1749:
1739:quadratic forms
1711:
1640:
1635:
1604:
1584:
1570:
1568:
1560:
1557:
1556:
1546:
1532:, PSL(2,
1467:
1453:
1451:
1450:
1446:
1387:
1386:
1381:
1375:
1374:
1369:
1359:
1358:
1320:
1317:
1316:
1306:
1292:
1256:Minkowski space
1213:
1145:
1095:identity matrix
1059:PSL(2, R)
959: and
956:
951:
917:
916:
911:
905:
904:
899:
889:
888:
887:
883:
872:
856:
854:
851:
850:
828:
806:
777:
776:
765:Abelian variety
758:Reductive group
746:
736:
735:
734:
733:
684:
676:
668:
660:
652:
625:Special unitary
536:
522:
521:
503:
502:
484:
482:
479:
478:
459:
457:
454:
453:
422:
420:
417:
416:
408:
407:
398:Discrete groups
387:
386:
342:Frobenius group
287:
274:
263:
256:Symmetric group
252:
236:
226:
225:
76:Normal subgroup
62:
42:
33:
28:
23:
22:
15:
12:
11:
5:
4757:
4747:
4746:
4741:
4736:
4731:
4726:
4712:
4711:
4697:
4684:
4670:
4644:
4640:
4636:
4631:
4626:
4623:
4608:
4562:(4): 337–342.
4547:
4526:
4500:(3): 568–640.
4484:
4483:
4468:
4441:
4440:
4438:
4435:
4433:
4432:
4427:
4422:
4420:Fuchsian group
4417:
4412:
4403:
4398:
4393:
4388:
4382:
4380:
4377:
4373:Harish-Chandra
4329:of type VIII.
4285:
4280:Main article:
4277:
4274:
4177:
4174:
4167:if and only if
4153:(PSL(2,
4150:
4128:
4127:
4116:
4113:
4109:
4105:
4102:
4099:
4095:
4092:
4089:
4085:
4082:
4078:
4074:
4071:
4068:
4064:
4061:
4057:
4054:
4050:
4046:
4043:
4040:
4036:
4033:
4029:
4026:
4023:
4018:
4014:
4010:
4006:
4003:
4000:
3996:
3993:
3948:
3921:
3882:
3878:
3874:
3870:
3867:
3864:
3828:
3824:
3820:
3816:
3813:
3810:
3774:
3770:
3766:
3762:
3759:
3756:
3717:
3713:
3709:
3705:
3702:
3699:
3675:representation
3656:
3652:
3648:
3644:
3641:
3638:
3602:
3598:
3594:
3590:
3587:
3584:
3508:
3505:
3501:
3500:
3489:
3485:
3480:
3476:
3473:
3466:
3459:
3455:
3451:
3448:
3445:
3442:
3439:
3436:
3431:
3425:
3422:
3420:
3417:
3416:
3413:
3410:
3408:
3405:
3404:
3402:
3396:
3392:
3388:
3377:
3366:
3362:
3358:
3355:
3352:
3345:
3338:
3334:
3330:
3327:
3324:
3321:
3318:
3315:
3310:
3302:
3299:
3295:
3291:
3289:
3286:
3285:
3282:
3279:
3277:
3274:
3273:
3271:
3265:
3261:
3257:
3246:
3235:
3232:
3229:
3226:
3223:
3220:
3217:
3213:
3208:
3204:
3201:
3194:
3187:
3183:
3179:
3176:
3173:
3170:
3167:
3164:
3159:
3153:
3150:
3147:
3144:
3142:
3139:
3136:
3133:
3132:
3129:
3126:
3123:
3120:
3117:
3115:
3112:
3109:
3106:
3105:
3103:
3097:
3093:
3089:
3065:
3061:
3057:
3054:
3051:
3016:
3013:
2992:
2989:
2967:
2964:
2961:
2958:
2955:
2951:
2941:
2938:
2934:
2930:
2928:
2925:
2922:
2921:
2917:
2900:of the torus.
2862:
2859:
2845:
2837:
2834:
2831:
2829:
2826:
2823:
2820:
2818:
2815:
2812:
2811:
2807:
2785:
2777:
2774:
2772:
2769:
2766:
2763:
2761:
2758:
2757:
2753:
2732:
2729:
2726:
2723:
2720:
2716:
2708:
2705:
2703:
2700:
2697:
2695:
2692:
2691:
2687:
2670:of the torus.
2649:limit rotation
2636:
2633:
2562:values on the
2551:
2548:
2466:, hyperbolic.
2455:
2452:
2449:
2429:
2426:
2423:
2403:
2400:
2397:
2362:conic sections
2358:
2357:
2333:
2330:
2326:
2322:
2319:
2316:
2312:
2309:
2304:
2292:
2268:
2265:
2261:
2257:
2254:
2251:
2247:
2244:
2239:
2227:
2203:
2200:
2196:
2192:
2189:
2186:
2182:
2179:
2174:
2158:
2157:
2146:
2141:
2135:
2132:
2127:
2123:
2119:
2116:
2112:
2109:
2103:
2100:
2097:
2094:
2090:
2087:
2080:
2077:
2065:and therefore
2063:
2062:
2051:
2047:
2043:
2039:
2035:
2031:
2028:
2025:
2021:
2018:
2013:
2007:
2003:
1987:) satisfy the
1983:∈ SL(2,
1979:of an element
1972:
1969:
1929:
1928:
1917:
1912:
1904:
1900:
1896:
1894:
1891:
1888:
1885:
1881:
1877:
1873:
1872:
1869:
1866:
1863:
1861:
1858:
1855:
1852:
1849:
1846:
1844:
1841:
1838:
1837:
1832:
1828:
1824:
1822:
1819:
1816:
1813:
1809:
1805:
1801:
1800:
1798:
1793:
1788:
1782:
1779:
1777:
1774:
1773:
1770:
1767:
1765:
1762:
1761:
1759:
1727:representation
1710:
1707:
1661:
1660:
1649:
1638:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1596:
1593:
1590:
1587:
1582:
1579:
1576:
1573:
1567:
1564:
1545:
1542:
1519:Riemann sphere
1511:
1510:
1499:
1495:
1491:
1485:
1479:
1476:
1473:
1470:
1465:
1462:
1459:
1456:
1449:
1444:
1438:
1435:
1432:
1429:
1426:
1420:
1417:
1414:
1411:
1408:
1405:
1399:
1391:
1385:
1382:
1380:
1377:
1376:
1373:
1370:
1368:
1365:
1364:
1362:
1357:
1354:
1348:
1345:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1291:
1288:
1272:
1271:
1248:
1233:
1219:
1180:SU(1, 1)
1159:that preserve
1144:
1141:
1112:, Mp(2,
1110:covering group
1087:
1086:
1042:) acts on the
1003:
1002:
991:
987:
983:
980:
977:
974:
971:
968:
965:
954:
950:
947:
944:
941:
938:
935:
932:
929:
926:
921:
915:
912:
910:
907:
906:
903:
900:
898:
895:
894:
892:
886:
882:
879:
875:
871:
868:
865:
826:
821:SL(2, R)
808:
807:
805:
804:
797:
790:
782:
779:
778:
775:
774:
772:Elliptic curve
768:
767:
761:
760:
754:
753:
747:
742:
741:
738:
737:
732:
731:
728:
725:
721:
717:
716:
715:
710:
708:Diffeomorphism
704:
703:
698:
693:
687:
686:
682:
678:
674:
670:
666:
662:
658:
654:
650:
645:
644:
633:
632:
621:
620:
609:
608:
597:
596:
585:
584:
573:
572:
565:Special linear
561:
560:
553:General linear
549:
548:
543:
537:
528:
527:
524:
523:
520:
519:
514:
509:
501:
500:
487:
475:
462:
449:
447:Modular groups
445:
444:
443:
438:
425:
409:
406:
405:
400:
394:
393:
392:
389:
388:
383:
382:
381:
380:
375:
370:
367:
361:
360:
354:
353:
352:
351:
345:
344:
338:
337:
332:
323:
322:
320:Hall's theorem
317:
315:Sylow theorems
311:
310:
305:
297:
296:
295:
294:
288:
283:
280:Dihedral group
276:
275:
270:
264:
259:
253:
248:
237:
232:
231:
228:
227:
222:
221:
220:
219:
214:
206:
205:
204:
203:
198:
193:
188:
183:
178:
173:
171:multiplicative
168:
163:
158:
153:
145:
144:
143:
142:
137:
129:
128:
120:
119:
118:
117:
115:Wreath product
112:
107:
102:
100:direct product
94:
92:Quotient group
86:
85:
84:
83:
78:
73:
63:
60:
59:
56:
55:
47:
46:
26:
9:
6:
4:
3:
2:
4756:
4745:
4742:
4740:
4737:
4735:
4732:
4730:
4727:
4725:
4722:
4721:
4719:
4708:
4704:
4700:
4698:0-691-08304-5
4694:
4690:
4685:
4681:
4677:
4673:
4671:0-387-96198-4
4667:
4663:
4659:
4655:
4629:
4609:
4605:
4601:
4596:
4591:
4587:
4583:
4578:
4573:
4569:
4565:
4561:
4557:
4553:
4548:
4544:
4540:
4536:
4532:
4527:
4523:
4519:
4515:
4511:
4507:
4503:
4499:
4495:
4494:
4488:
4487:
4479:
4475:
4471:
4465:
4461:
4457:
4453:
4446:
4442:
4431:
4428:
4426:
4423:
4421:
4418:
4416:
4413:
4411:
4409:
4404:
4402:
4401:Modular group
4399:
4397:
4394:
4392:
4389:
4387:
4384:
4383:
4376:
4374:
4370:
4366:
4362:
4358:
4353:
4351:
4347:
4343:
4339:
4335:
4330:
4328:
4324:
4320:
4316:
4312:
4308:
4304:
4300:
4294:
4293:
4291:
4273:
4271:
4267:
4263:
4259:
4255:
4250:
4248:
4244:
4240:
4236:
4233:
4228:
4226:
4223:PSL(2,
4222:
4221:modular group
4218:
4217:Frieze groups
4214:
4210:
4207:) are called
4206:
4201:
4199:
4195:
4192:PSL(2,
4191:
4187:
4183:
4173:
4171:
4168:
4164:
4160:
4156:
4148:
4144:
4141:
4137:
4133:
4114:
4103:
4100:
4072:
4069:
4044:
4041:
4024:
4004:
4001:
3980:
3979:
3978:
3975:
3973:
3969:
3965:
3961:
3956:
3954:
3947:
3943:
3939:
3931:
3930:modular group
3927:
3920:
3917:
3912:
3908:
3906:
3902:
3898:
3897:circle bundle
3868:
3865:
3844:
3814:
3811:
3790:
3760:
3757:
3737:
3733:
3703:
3700:
3678:
3676:
3672:
3642:
3639:
3618:
3588:
3585:
3564:
3560:
3557:
3553:
3548:
3546:
3542:
3538:
3534:
3530:
3526:
3525:circle bundle
3522:
3518:
3514:
3504:
3487:
3483:
3474:
3471:
3449:
3446:
3440:
3437:
3434:
3429:
3423:
3418:
3411:
3406:
3400:
3394:
3390:
3378:
3364:
3360:
3356:
3353:
3350:
3328:
3325:
3319:
3316:
3313:
3308:
3300:
3297:
3293:
3287:
3280:
3275:
3269:
3263:
3259:
3247:
3233:
3227:
3221:
3218:
3215:
3211:
3202:
3199:
3177:
3174:
3168:
3165:
3162:
3157:
3151:
3148:
3145:
3140:
3137:
3134:
3127:
3124:
3121:
3118:
3113:
3110:
3107:
3101:
3095:
3091:
3079:
3078:
3077:
3055:
3052:
3034:
3030:
3026:
3022:
3012:
3008:
3006:
3002:
2998:
2988:
2985:
2981:
2962:
2959:
2953:
2949:
2939:
2936:
2932:
2923:
2915:
2906:
2901:
2899:
2895:
2894:modular group
2890:
2888:
2884:
2883:Lorentz boost
2880:
2876:
2872:
2868:
2858:
2843:
2835:
2832:
2824:
2821:
2816:
2813:
2805:
2783:
2775:
2767:
2764:
2759:
2751:
2727:
2724:
2718:
2714:
2706:
2698:
2693:
2685:
2676:
2671:
2669:
2665:
2664:modular group
2660:
2658:
2654:
2653:null rotation
2650:
2646:
2642:
2641:shear mapping
2632:
2630:
2626:
2621:
2619:
2615:
2611:
2606:
2604:
2600:
2596:
2595:root of unity
2592:
2588:
2587:modular group
2583:
2581:
2577:
2573:
2569:
2565:
2561:
2557:
2547:
2545:
2541:
2536:
2534:
2530:
2526:
2522:
2517:
2515:
2512:
2510:
2505:
2500:
2498:
2494:
2490:
2486:
2482:
2478:
2474:
2472:
2467:
2453:
2450:
2447:
2440:, parabolic;
2427:
2424:
2421:
2401:
2398:
2395:
2387:
2367:
2363:
2355:
2351:
2347:
2331:
2328:
2317:
2293:
2290:
2289:shear mapping
2286:
2282:
2266:
2263:
2252:
2228:
2225:
2221:
2217:
2201:
2198:
2187:
2163:
2162:
2161:
2144:
2139:
2133:
2130:
2125:
2117:
2101:
2095:
2078:
2075:
2068:
2067:
2066:
2049:
2045:
2041:
2037:
2033:
2026:
2011:
2005:
2001:
1993:
1992:
1991:
1990:
1986:
1982:
1978:
1968:
1966:
1962:
1958:
1954:
1950:
1949:Lorentz group
1946:
1942:
1938:
1934:
1915:
1910:
1902:
1898:
1892:
1889:
1886:
1879:
1875:
1867:
1864:
1859:
1856:
1853:
1850:
1847:
1842:
1839:
1830:
1826:
1820:
1817:
1814:
1807:
1803:
1796:
1786:
1780:
1775:
1768:
1763:
1757:
1748:
1747:
1746:
1744:
1740:
1736:
1732:
1728:
1724:
1720:
1716:
1706:
1704:
1700:
1696:
1691:
1689:
1685:
1681:
1676:
1674:
1670:
1666:
1647:
1632:
1629:
1626:
1623:
1620:
1617:
1614:
1611:
1594:
1591:
1588:
1585:
1580:
1577:
1574:
1571:
1562:
1555:
1554:
1553:
1551:
1541:
1539:
1535:
1531:
1526:
1524:
1520:
1516:
1497:
1493:
1489:
1483:
1477:
1474:
1471:
1468:
1463:
1460:
1457:
1454:
1447:
1442:
1433:
1430:
1427:
1424:
1418:
1415:
1412:
1409:
1406:
1397:
1389:
1383:
1378:
1371:
1366:
1360:
1352:
1346:
1343:
1331:
1328:
1325:
1315:
1314:
1313:
1309:
1305:
1301:
1297:
1287:
1285:
1281:
1277:
1269:
1265:
1261:
1257:
1253:
1252:Lorentz group
1249:
1246:
1242:
1238:
1234:
1231:
1227:
1226:automorphisms
1224:
1220:
1216:
1212:
1208:
1204:
1200:
1199:
1198:
1196:
1191:
1189:
1185:
1184:coquaternions
1181:
1177:
1173:
1169:
1165:
1162:
1158:
1154:
1150:
1140:
1138:
1137:modular group
1134:
1129:
1127:
1123:
1119:
1115:
1111:
1106:
1104:
1101:PSL(2,
1100:
1099:modular group
1096:
1092:
1084:
1080:
1076:
1073:PSL(2,
1072:
1071:
1070:
1068:
1064:
1060:
1057:
1053:
1049:
1045:
1041:
1036:
1034:
1030:
1026:
1022:
1018:
1014:
1011:
1008:
989:
985:
981:
978:
975:
972:
969:
966:
963:
948:
945:
942:
939:
936:
933:
930:
927:
924:
919:
913:
908:
901:
896:
890:
884:
880:
869:
866:
849:
848:
847:
845:
841:
838:
834:
830:
822:
819:
815:
803:
798:
796:
791:
789:
784:
783:
781:
780:
773:
770:
769:
766:
763:
762:
759:
756:
755:
752:
749:
748:
745:
740:
739:
729:
726:
723:
722:
720:
714:
711:
709:
706:
705:
702:
699:
697:
694:
692:
689:
688:
685:
679:
677:
671:
669:
663:
661:
655:
653:
647:
646:
642:
638:
635:
634:
630:
626:
623:
622:
618:
614:
611:
610:
606:
602:
599:
598:
594:
590:
587:
586:
582:
578:
575:
574:
570:
566:
563:
562:
558:
554:
551:
550:
547:
544:
542:
539:
538:
535:
531:
526:
525:
518:
515:
513:
510:
508:
505:
504:
476:
451:
450:
448:
442:
439:
414:
411:
410:
404:
401:
399:
396:
395:
391:
390:
379:
376:
374:
371:
368:
365:
364:
363:
362:
359:
356:
355:
350:
347:
346:
343:
340:
339:
336:
333:
331:
329:
325:
324:
321:
318:
316:
313:
312:
309:
306:
304:
301:
300:
299:
298:
292:
289:
286:
281:
278:
277:
273:
268:
265:
262:
257:
254:
251:
246:
243:
242:
241:
240:
235:
234:Finite groups
230:
229:
218:
215:
213:
210:
209:
208:
207:
202:
199:
197:
194:
192:
189:
187:
184:
182:
179:
177:
174:
172:
169:
167:
164:
162:
159:
157:
154:
152:
149:
148:
147:
146:
141:
138:
136:
133:
132:
131:
130:
127:
126:
122:
121:
116:
113:
111:
108:
106:
103:
101:
98:
95:
93:
90:
89:
88:
87:
82:
79:
77:
74:
72:
69:
68:
67:
66:
61:Basic notions
58:
57:
53:
49:
48:
45:
40:
36:
32:
31:
19:
4724:Group theory
4688:
4612:
4559:
4555:
4534:
4530:
4497:
4491:
4460:10.1142/p835
4451:
4445:
4407:
4406:SL(2,
4386:Linear group
4371:(1947), and
4356:
4354:
4345:
4341:
4333:
4331:
4318:
4314:
4306:
4298:
4297:SL(2,
4296:
4289:
4283:
4261:
4257:
4251:
4246:
4242:
4232:circle group
4229:
4224:
4204:
4202:
4193:
4185:
4179:
4169:
4162:
4154:
4146:
4142:
4139:
4135:
4131:
4129:
3976:
3971:
3967:
3959:
3957:
3945:
3937:
3935:
3918:
3679:
3619:. That is,
3617:matrix group
3558:
3556:cyclic group
3551:
3549:
3540:
3536:
3516:
3510:
3502:
3032:
3028:
3024:
3018:
3009:
3004:
3000:
2994:
2904:
2902:
2891:
2877:) acts as a
2874:
2864:
2674:
2672:
2661:
2647:) acts as a
2644:
2638:
2622:
2613:
2609:
2607:
2590:
2584:
2571:
2553:
2537:
2528:
2524:
2520:
2518:
2513:
2507:
2503:
2501:
2492:
2484:
2480:
2475:
2470:
2468:
2414:, elliptic;
2385:
2366:eccentricity
2359:
2349:
2345:
2284:
2280:
2219:
2215:
2159:
2064:
1984:
1980:
1974:
1960:
1952:
1944:
1936:
1933:Killing form
1930:
1742:
1734:
1730:
1718:
1714:
1712:
1692:
1677:
1668:
1662:
1549:
1547:
1536:) expresses
1533:
1527:
1514:
1512:
1307:
1300:homographies
1295:
1293:
1290:Homographies
1283:
1279:
1275:
1273:
1263:
1239:-preserving
1214:
1205:-preserving
1194:
1192:
1187:
1175:
1174:Sp(2,
1156:
1148:
1147:SL(2,
1146:
1143:Descriptions
1132:
1130:
1121:
1113:
1107:
1102:
1090:
1088:
1082:
1078:
1074:
1066:
1058:
1052:group action
1039:
1038:SL(2,
1037:
1004:
824:
820:
811:
640:
628:
616:
604:
592:
580:
568:
556:
327:
284:
271:
260:
249:
245:Cyclic group
123:
110:Free product
81:Group action
44:Group theory
39:Group theory
38:
4744:3-manifolds
4430:Anosov flow
4369:V. Bargmann
4311:Lie algebra
3942:braid group
3916:braid group
2879:translation
2867:eigenvalues
2668:Dehn twists
2620:in PSL(2).
2618:involutions
2564:unit circle
2556:eigenvalues
2350:hyperbolic,
1977:eigenvalues
1723:conjugation
1237:orientation
1203:orientation
1139:, however.
1010:non-compact
844:determinant
814:mathematics
530:Topological
369:alternating
4729:Lie groups
4718:Categories
4437:References
3736:3-manifold
3669:admits no
3565:, denoted
2544:closed set
2529:subgroups:
2348:is called
2285:parabolic,
2283:is called
2218:is called
1947:) and the
1680:isometries
1270:Spin(2,1).
1268:spin group
1241:isometries
1168:isomorphic
637:Symplectic
577:Orthogonal
534:Lie groups
441:Free group
166:continuous
105:Direct sum
4537:: 93–94.
4323:traceless
4172:is even.
4084:→
4056:→
4028:→
4025:⋯
4022:→
4017:¯
3940:) is the
3881:¯
3827:¯
3773:¯
3716:¯
3655:¯
3601:¯
3475:∈
3435:∈
3314:∈
3298:−
3216:≅
3203:∈
3200:θ
3163:∈
3152:θ
3149:
3141:θ
3138:
3128:θ
3125:
3119:−
3114:θ
3111:
2960:±
2954:×
2937:−
2933:λ
2924:λ
2833:−
2822:±
2814:−
2765:±
2725:±
2719:×
2699:λ
2560:conjugate
2448:ϵ
2422:ϵ
2396:ϵ
2352:and is a
2287:and is a
2220:elliptic,
2131:−
2102:±
2076:λ
2034:λ
2012:−
2002:λ
1963:) on the
1941:signature
1792:↦
1633:∈
1566:↦
1338:↦
1230:unit disc
1223:conformal
1017:Lie group
1007:connected
970:−
949:∈
925::
701:Conformal
589:Euclidean
196:nilpotent
4604:16589101
4379:See also
4375:(1952).
4367:(1946),
4309:). The
4270:topology
4190:quotient
3901:orbifold
3671:faithful
3003:,
2576:rotation
2568:rotation
2540:open set
2525:subsets,
2224:rotation
1166:. It is
1161:oriented
1056:quotient
1025:topology
1021:geometry
1005:It is a
840:matrices
696:Poincaré
541:Solenoid
413:Integers
403:Lattices
378:sporadic
373:Lie type
201:solvable
191:dihedral
176:additive
161:infinite
71:Subgroup
4707:1435975
4680:0803508
4595:1063558
4586:0047055
4564:Bibcode
4543:0017282
4522:0021942
4514:1969129
4478:2977041
4365:Naimark
4361:Gelfand
3928:of the
3924:is the
3561:. The
3545:spinors
3035:. For
2896:act as
2666:act as
2382:
2370:
2344:, then
2279:, then
2214:, then
1682:of the
1302:on the
1243:of the
1228:of the
1209:of the
1170:to the
1124:) as a
1033:physics
831:is the
691:Lorentz
613:Unitary
512:Lattice
452:PSL(2,
186:abelian
97:(Semi-)
18:PSL2(R)
4705:
4695:
4678:
4668:
4602:
4592:
4584:
4541:
4520:
4512:
4476:
4466:
4198:simple
4182:center
3734:, the
3732:metric
3469:
3461:
3348:
3340:
3197:
3189:
2984:arcosh
2978:; the
2629:arccos
1939:) has
1487:
1440:
1422:
1401:
1395:
1350:
1298:) are
1217:∪ {∞}.
1089:where
1050:. The
1031:, and
1013:simple
816:, the
546:Circle
477:SL(2,
366:cyclic
330:-group
181:cyclic
156:finite
151:simple
135:kernel
4510:JSTOR
4237:is a
4235:SO(2)
4196:) is
4145:<
4138:, as
3962:), a
3511:As a
2603:torus
2599:order
1955:) on
1721:) by
1310:∪ {∞}
1116:), a
1065:over
1015:real
846:one:
842:with
833:group
730:Sp(∞)
727:SU(∞)
140:image
4693:ISBN
4666:ISBN
4600:PMID
4464:ISBN
4363:and
4252:The
4230:The
4215:and
4180:The
3914:The
3354:>
3019:The
2865:The
2554:The
2527:not
2451:>
2399:<
2329:>
2199:<
1975:The
1931:The
1697:and
1695:dual
1164:area
837:real
724:O(∞)
713:Loop
532:and
4658:doi
4590:PMC
4572:doi
4502:doi
4456:doi
4249:).
4149:≅ π
3974:).
3907:).
3903:(a
3146:cos
3135:sin
3122:sin
3108:cos
2995:By
2885:on
2655:of
2516:).
2487:) (
2364:by
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