Projective Linear Groups as Maximal Symmetry Groups

Abstract

Abstract.A maximal symmetry group is a group of isomorphisms of a threedimensional hyperbolic manifold of maximal order in relation to the volume of the manifold. In this paper we determine all maximal symmetry groups of the types PSL(2, q) and PGL (2, q). Depending on the prime p there are one or two such groups with q = p k and k always equals 1, 2 or 4.2000 Mathematics Subject Classification. 20B25, 20G40.

Hyperbolic manifolds and tessellations of type {3,5,3} associated with L_2(q)

“…In §4 we give a criterion to determine which of these quotients arises. This is more general than that given by Paoluzzi in [24], where only the case q ≡ 1 mod (10) is considered, and is rather simpler than that given by Torstensson in [28]. Theorem 1.2 does not extend to all finite simple quotients of ∆: for instance, ∆ has four normal subgroups with quotient isomorphic to the alternating group A 25 , and only two of these are normal in Ω + .…”

“…The case p = 2 can be dealt with in a similar way, using (3.2) instead of (3.1), but we will omit the details since the result follows immediately from Torstensson's work. She has shown in [28] that Ω + (denoted there by Γ) has a normal subgroup N with Ω + /N ∼ = L 2 (2 4 ); then N ∩ ∆ is the subgroup K in Theorem 1.1(a), and this is normal in Ω + with quotient…”

“…Paoluzzi [24, Theorem 1] has determined which groups L 2 (q) are quotients of ∆ (her tetrahedral group T 5,2 ), and has also obtained partial results on quotients of Ω + and Γ (her T and C 5,2 ). Torstensson [28] has extended these results by determining all the quotients of Ω + (her group Γ) of type L 2 (q) or P GL 2 (q). Conder, Martin and Torstensson [2] have also considered the groups L 2 (q), and also alternating and symmetric groups, as quotients.…”

“…In §11 we give tables of our results for all primes p ≤ 251, extending those in [24] and [28]; these were obtained by using GAP to check the relevant conditions in finite fields.…”

Congruence subgroups of the minimal covolume arithmetic Kleinian group

“…By the recent work of Gehring and Martin (for more details see [4] as well as the introduction of [5] and [10]) the group is the lattice with the smallest covolume among all lattices in SL 2 (C). By this result, + is the right analog of the triangle group (2,3,7) for the 3-dimensional hyperbolic space H 3 .…”