Morse Theory and Conjugacy Classes of Finite Subgroups

We construct a CAT(0) group containing a finitely presented subgroup with infinitely many conjugacy classes of finiteorder elements. Unlike previous examples (which were based on right-angled Artin groups) our ambient CAT(0) group does not contain any rank 3 free abelian subgroups. We also construct examples of groups of type Fn inside mapping class groups, Aut(Fk), and Out(Fk) which have infinitely many conjugacy classes of finite-order elements. Hyperbolic groups contain only finitely many conjugacy classes of finite subgroups (see [2, 5, 7]). Several other classes of groups share this property, including CAT(0) groups [7, Corollary II.2.8], mapping class groups [6], Aut(Fk), Out(Fk) [8], and arithmetic groups [3]. Building on work of Grunewald and Platonov [11], Bridson [6] showed that for any n, there is a subgroup of SL(2n + 2,Z) of type Fn that has infinitely many conjugacy classes of elements of order 4. In [10], Feighn and Mess constructed finite extensions of (F2) n containing subgroups of type Fn−1 with infinitely many conjugacy classes of elements of order 2. Their examples were realized as subgroups of the isometry group of (H) = H × · · · ×H. These examples were generalized considerably and were set in the context of right-angled Artin groups by Leary and Nucinkis [12]. In this note we describe a model situation where the Morse theory argument of [12] can be applied. It includes the right-angled Artin group setting from [12], but it can also be used when the ambient group is not an Artin group. We apply it to the case of a hyperbolic group in Theorem 2.1 and to the case of a virtual direct product of hyperbolic groups in Theorem 2.2. In addition, we produce subgroups of mapping class groups, Aut(Fk), and Out(Fk) with infinitely many conjugacy classes of finite-order elements by finding natural realizations of finite extensions of (F2) n in these groups (Theorems 3.1, 3.2, and 3.3). For mapping class groups this solves Problem 3.10 in [9]. Date: September 21, 2007. N. Brady was partially supported by NSF grant no. DMS-0505707