Train Tracks, Orbigraphs and CAT(0) Free-by-cyclic Groups

Rylee Alanza Lyman

Published 2019-09-06Version 1

Gersten gave an example of a polynomially-growing automorphism of $F_3$ whose mapping torus $F_3 \rtimes \mathbb{Z}$ cannot act properly by semi-simple isometries on a CAT(0) metric space. By contrast, we show that if $\Phi$ is a polynomially-growing automorphism belonging to one of several related groups, there exists $k > 0$ such that the mapping torus of $\Phi^k$ acts properly and cocompactly on a CAT(0) metric space. This $k$ can often be bounded uniformly. Our results apply to automorphisms of a free product of $n$ copies of a finite group $A$, as well as to palindromic and symmetric automorphisms of a free group of finite rank. Of independent interest, a key tool in our proof is the construction of relative train track maps on orbigraphs, certain graphs of groups thought of as orbi-spaces.