When loxodromics are pseudo-Anosovs on witnesses

TL;DR

This paper characterizes loxodromic elements in certain groups as those that are pseudo-Anosov on witnesses, extending Masur and Minsky's result to broader contexts involving multiarc, curve, and disk graphs.

Contribution

It generalizes the characterization of loxodromic elements as pseudo-Anosovs to subgroups acting on multiarc, curve, and disk graphs, linking algebraic and geometric properties.

Findings

Loxodromic elements correspond to pseudo-Anosovs on witnesses.

The result extends Masur and Minsky's theorem to new group actions.

Pure powers of these elements are pseudo-Anosovs on witnesses.

Abstract

In this paper, we prove that for subgroups acting on admissible multiarc and curve graphs and for the handlebody group acting on the disk graph, the loxodromic elements are exactly those for which some pure power is a pseudo-Anosov on a witness. This generalizes the result of Masur and Minsky that the elements of the mapping class group that act loxodromically on the curve graph are the pseudo-Anosov elements.