Abstract

The main part of my talk will present some results obtained with Jerome Los: suppose we are given a finite graph $X$ and a subgroup $G$ of its automorphism group. Under what conditions can we find a surface $S$ (with boundary) together with a (finite) subgroup $H$ of its homeomorphism group, such that $H$ has an invariant subgraph homeomorphic to $X$ and the restriction of $H$ to this subgraph is equivalent to $G$? We give necessary and sufficient conditions for this to be possible, based primarily on the combinatorics of the action of stabilizer subgroups of vertices. In particular, conditions can be given which insure that the surface $S$ can be chosen to be orientable, and further ones that insure that the homeomorphisms in $H$ can be chosen to preserve orientation.
These results are a few years old, but are only now about to appear, in Topology. About a year ago, in a PhD thesis at Tufts, Christopher Thomas used these results to obtain necessary, and often sufficient, conditions for a finite subgroup of $Out(F$_n) to be realizable as the action of a finite group of homeomorphisms on the fundamental group of a punctured surface; I will explain his results as well.