Big Flip Graphs and Their Automorphism Groups

TL;DR

This paper investigates the automorphism groups of flip graphs for infinite-type surfaces, revealing that the extended mapping class group is a proper subgroup, thus challenging Ivanov's metaconjecture in this context.

Contribution

It demonstrates that the automorphism group of the flip graph for infinite-type surfaces is larger than the extended mapping class group, providing a counterexample to the extension of Ivanov's metaconjecture.

Findings

Extended mapping class group is a proper subgroup of the flip graph automorphism group.

Ivanov's metaconjecture does not hold for infinite-type surfaces.

Automorphism groups of flip graphs are strictly larger than the mapping class groups.

Abstract

In this paper, we study the relationship between the mapping class group of an infinite-type surface and the simultaneous flip graph, a variant of the flip graph for infinite-type surfaces defined by Fossas and Parlier. We show that the extended mapping class group is isomorphic to a proper subgroup of the automorphism group of the flip graph, unlike in the finite-type case. This shows that Ivanov's metaconjecture, which states that any "sufficiently rich" object associated to a finite-type surface has the extended mapping class group as its automorphism group, does not extend to simultaneous flip graphs of infinite-type surfaces.