Morse theory and conjugacy classes of finite subgroups

TL;DR

This paper constructs new examples of CAT(0) groups and other groups with infinitely many conjugacy classes of finite-order elements, highlighting novel geometric and algebraic properties without relying on rank 3 free abelian subgroups.

Contribution

It introduces constructions of CAT(0) groups and groups within mapping class groups with infinitely many conjugacy classes of finite-order elements, avoiding previous structures like rank 3 free abelian subgroups.

Findings

Constructed a CAT(0) group with a finitely presented subgroup with infinitely many conjugacy classes of finite-order elements

Provided examples of groups of type F_n with infinitely many conjugacy classes of finite-order elements within mapping class groups, Aut(F), and Out(F)

Demonstrated new geometric group theory phenomena beyond prior examples based on right-angled Artin groups

Abstract

We construct a CAT(0) group containing a finitely presented subgroup with infinitely many conjugacy classes of finite-order 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 F_n inside mapping class groups, Aut(F), and Out(F) which have infinitely many conjugacy classes of finite-order elements.