Homomorphisms into mapping class groups. An addendum

TL;DR

This paper extends previous work by showing that groups with many non-conjugate homomorphisms into mapping class groups have actions on trees, revealing structural properties of such groups.

Contribution

It demonstrates, using new methods and an auxiliary result, that finitely generated groups with infinitely many homomorphisms into mapping class groups virtually act on trees.

Findings

Finitely generated groups with infinitely many homomorphisms act on $ ext{R}$-trees.

Finitely presented groups with such homomorphisms act on simplicial trees.

Utilizes methods from previous work and an auxiliary result by Bestvina-Bromberg-Fujiwara.

Abstract

This is an addendum to arXiv: 0810.5376. We show, using our methods and an auxiliary result of Bestvina-Bromberg-Fujiwara, that a finitely generated group with infinitely many pairwise non-conjugate homomorphisms to a mapping class group virtually acts non-trivially on an $\R$-tree, and, if it is finitely presented, it virtually acts non-trivially on a simplicial tree