Alternating maps on Hatcher-Thurston graphs

TL;DR

This paper investigates edge-preserving alternating maps between Hatcher-Thurston graphs of surfaces, establishing genus inequalities and conditions under which such maps are induced by homeomorphisms, revealing structural constraints of these maps.

Contribution

It proves genus inequalities for surfaces under alternating maps and characterizes when these maps are induced by homeomorphisms, extending understanding of surface graph automorphisms.

Findings

$g_1 \,\leq\, g_2$ for the surfaces involved

Existence of a multicurve of size $g_2 - g_1$ in the image

When $n_1=0$ and $g_1=g_2$, the map is induced by a homeomorphism

Abstract

Let $S_{1}$ and $S_{2}$ be connected orientable surfaces of genus $g_{1}, g_{2} \geq 3$, $n_{1},n_{2} \geq 0$ punctures, and empty boundary. Let also $\varphi: \mathcal{HT}(S_{1}) \rightarrow \mathcal{HT}(S_{2})$ be an edge-preserving alternating map between their Hatcher-Thurston graphs. We prove that $g_{1} \leq g_{2}$ and that there is also a multicurve of cardinality $g_{2} - g_{1}$ contained in every element of the image. We also prove that if $n_{1} = 0$ and $g_{1} = g_{2}$, then the map $\widetilde{\varphi}$ obtained by filling the punctures of $S_{2}$, is induced by a homeomorphism of $S_{1}$.