Automorphisms of the compression body graph

TL;DR

This paper proves that the automorphism group of the compression body graph for a closed, orientable surface of genus at least 2 is isomorphic to the surface's mapping class group, revealing a deep symmetry connection.

Contribution

It establishes that the automorphism group of the compression body graph coincides with the mapping class group for surfaces of genus at least 2, linking graph symmetries to surface homeomorphisms.

Findings

Automorphism group of the compression body graph is the mapping class group.

Vertices represent compression bodies with boundary surface S.

Edges indicate containment relations between compression bodies.

Abstract

When $S$ is a closed, orientable surface with genus $g(S) \geq 2$, we show that the automorphism group of the compression body graph $\mathcal{CB}(S)$ is the mapping class group. Here, vertices are compression bodies with exterior boundary $S$, and edges connect pairs of compression bodies where one contains the other.