Reconstructing simplicial group actions.

Abstract: Several simplicial complexes which are important from a combinatorics, geometry or representation theory have enormous symmetry groups. Associated to the action of a group G on a simplicial complex X, one has the (typically much smaller and simpler) quotient space X/G. This quotient does not, by itself, carry enough structure to reconstruct either X or the G-action. The purpose of this talk is to describe what additional algebraic data is needed in order to perform such a reconstruction. We will decorate the simplices of X/G with subgroups of G in a coherent manner to obtain a “simplicial complex of subgroups of G”. Knowledge of these overlaid subgroups, and how they embed in G relative to one another, provides precise algorithmic instructions for how to unfold X/G so that X and the G-action are correctly recovered.