The Hamiltonian surface of $\Aut(F_2)$

TL;DR

This paper investigates the geometric structure of the automorphism group of a free group on two generators, revealing it contains exactly two Hamiltonian surfaces that traverse all vertices and edges exactly once.

Contribution

It proves that the CAT(0) space associated with Aut(F_2) contains exactly two Hamiltonian surfaces, providing new insights into its geometric and combinatorial properties.

Findings

Exactly two Hamiltonian surfaces in the CAT(0) space

Hamiltonian surfaces visit every vertex and edge exactly once

Enhanced understanding of the geometric structure of Aut(F_2)

Abstract

The group of automorphisms of the free group on two generators is known to act geometrically, in an essentially unique way, on a 2-dimensional CAT(0) space X. We prove that X contains precisely two Hamiltonian surfaces. By this we mean a surface in X which visits every vertex and every edge precisely once.