Finite Rigid Sets in Curve Complexes of Non-Orientable Surfaces

TL;DR

This paper identifies finite rigid sets within the curve complexes of non-orientable surfaces, providing insights into their structure and symmetries, which are crucial for understanding surface homeomorphisms.

Contribution

It introduces the first finite rigid sets in the curve complexes of non-orientable surfaces with genus and holes, expanding the understanding of their combinatorial properties.

Findings

Finite rigid sets exist in the curve complexes of non-orientable surfaces.

These sets are characterized for surfaces with genus g and n holes, excluding g+n=4.

The results aid in classifying surface homeomorphisms via combinatorial data.

Abstract

A rigid set in a curve complex of a surface is a subcomplex such that every locally injective simplicial map from the set into the curve complex is induced by a homeomorphism of the surface. In this paper, we find finite rigid sets in the curve complexes of connected non-orientable surfaces of genus $g$ with $n$ holes for $g+n \neq 4$.