Finiteness properties for subgroups of GL $(n,\mathbb Z)$

Abstract

We construct finitely presented subgroups of GL(n, Z) that have infinitely many conjugacy classes of finite subgroups. This answers a question of Grunewald and Platonov. We suggest a variation on their question. Mathematics Subject Classification (1991): 20G30

Morse theory and conjugacy classes of finite subgroups

“…Remark 1.2. Proposition 1.1 generalizes Lemma 2 in [5] where Q is an infinite abelian group, and the hypothesis on the index of the centralizers is ensured by requiring |ϕ(Cent G (σ))| < ∞.…”

On the subgroups of right-angled Artin groups and mapping class groups

“…(Mapping class groups themselves have only finitely many conjugacy classes of finite subgroups -see the appendix to [8], also [26].) This can be settled by combining Proposition 5.1 with the constructions used in [8] to provide the first examples of finitely presented subgroups of SL(n, Z) that have infinitely many conjugacy classes of finite subgroups, as we shall explain. We give two explicit examples of subgroups as described in the preceding proposition.…”

“…The hyperbolic orbifold H 2 /T n is a torus with a cone point of index n, corresponding to the fixed point of [a, b]. For each epimorphism T n → Z, the kernel of the induced map from D = T n × T n × T n × T n onto Z is finitely presented and has infinitely many conjugacy classes of elements of order n (see [8] Proposition 4). D has a subgroup of finite index that is a direct product of four copies of the fundamental group of a closed hyperbolic surface.…”

“…It is proved on page 630 of [8] that K n is finitely presented, and among the cyclic subgroups ( x −1 γx, γ, . . .…”

Новые Свойства Арифметических Групп

“…В [19] теорема 4.1 использована для построения конечно представимой под-группы в GL 6 (Z), которая содержит бесконечное число классов сопряженности элементов порядка 4.…”