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.Comment: 10 pages, 4 figure

Brady, Noel

Clay, Matt

Dani, Pallavi

English

arXiv

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 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 (H2)n = H2 ×  ·  ·  ·  × H2. Using the techniques of Morse functions on CAT(0) cubical complexes, Leary and Nucinkis [13], constructed a very general class of examples related to right-angled Artin groups. Further examples appeared in [12]. In this note we describe a model situation where the Morse theory argument of [13] can be applied. It includes the right-angled Artin group setting from [13], 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 finit

Noel Brady

Matt Clay

Pallavi Dani

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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 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 Fn inside mapping class groups, Aut(double-struck Fk), and Out(double-struck Fk) which have infinitely many conjugacy classes of finite-order elements. © 2008 Springer Science+Business Media B.V

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Morse theory and conjugacy classes of finite subgroups

https://www.math.lsu.edu/~pdani/research/conj-classes.pdf

Morse theory and conjugacy classes of finite subgroups

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