Edge Preserving Maps of the Nonseparating Curve Graphs, Curve Graphs and Rectangle Preserving Maps of the Hatcher-Thurston Graphs
Elmas Irmak

TL;DR
This paper demonstrates that certain edge-preserving and rectangle-preserving maps on various curve and Hatcher-Thurston graphs of a surface are induced by homeomorphisms, establishing a strong link between graph symmetries and surface homeomorphisms.
Contribution
It proves that edge-preserving maps on nonseparating and curve graphs, as well as rectangle-preserving maps on Hatcher-Thurston graphs, correspond to homeomorphisms of the surface, with uniqueness up to isotopy.
Findings
Edge-preserving maps on nonseparating curve graphs are induced by surface homeomorphisms.
Edge-preserving maps on curve graphs are induced by surface homeomorphisms.
Rectangle-preserving maps on Hatcher-Thurston graphs are induced by surface homeomorphisms.
Abstract
Let be a compact, connected, orientable surface of genus with boundary components with , . Let be the nonseparating curve graph, be the curve graph and be the Hatcher-Thurston graph of . We prove that if is an edge-preserving map, then is induced by a homeomorphism of . We prove that if is an edge-preserving map, then is induced by a homeomorphism of . We prove that if is closed and is a rectangle preserving map, then is induced by a homeomorphism of . We also prove that these homeomorphisms are unique up to isotopy when .
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
