Morse theory and conjugacy classes of finite subgroups

Abstract

Abstract. We construct a hyperbolic group containing a finitely presented subgroup, which has infinitely many conjugacy classes of finite-order elements.We also use a version of Morse theory with high dimensional horizontal cells and use handle cancellation arguments to produce other examples of subgroups of CAT(0) groups with infinitely many conjugacy classes of finite-order elements.

Morse theory and conjugacy classes of finite subgroups II

“…This paper is a continuation of our earlier work in [3]. In that paper we showed how the construction of Leary-Nucinkis [8] fits into a more general framework than that of right angled Artin groups.…”

“…Let σ ∈ G have finite order and the property that f (Fix(σ)) ⊂ R is compact. This generalizes our model situation from [3] where we required that σ had an isolated fixed point. Then g ∈ Cent G (σ) implies that g acts on Fix(σ) and therefore by ϕ-equivariance of f , ϕ(g) acts on f (Fix(σ)).…”

“…This allows us to see (in Section 2) that the original examples of Feighn-Mess [7] fit into the same general framework as the examples of Leary-Nucinkis [8]. This addresses a remark we made after Example 1.2 of [3].…”

Vanishing of ℓp-cohomology and transportation cost

“…As noted by Kappos [12], the previous theorem covers virtually nilpotent groups. For more amusing examples of groups with this property, see [3].…”

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

“…We give two explicit examples of subgroups as described in the preceding proposition. Earlier examples are due to Brady et al [4].…”