Alternating maps on Hatcher-Thurston graphs

Abstract

Let $S_1$ and $S_2$ be connected orientable surfaces of genus $g_1,g_2\ge 3$, $n_1,n_2\ge 0$ punctures, and empty boundary. Let also $\phi :\mathcal{H}\mathcal{T}(S_1)\to \mathcal{H}\mathcal{T}(S_2)$ be an edge-preserving alternating map between their Hatcher-Thurston graphs. We prove that $g_1\le g_2$ 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 $\tilde{\phi }$ obtained by filling the punctures of $S_2$, is induced by a homeomorphism of $S_1$.

16 pages, 8 figures

References in corpus (2)

• An elementary approach to the mapping class group of a surface
• Edge-preserving maps of curve graphs

Cited by in corpus (1)

• Edge Preserving Maps of the Curve Graphs in Low Genus