Big mapping class groups: an overview

Abstract

We survey recent developments on mapping class groups of surfaces of infinite topological type.

Big mapping class groups and the co-Hopfian property

“…In the case when S has finite type, Map(S) is well-known to be finitely presented. If, on the contrary, S has infinite type, then Map(S) becomes an uncountable, totally disconnected, non-locally compact topological group with respect to the quotient topology stemming from the compact-open topology on the homeomorphism group of S. We refer the reader to Section 2 for expanded definitions, and to the recent survey [5] for a detailed treatment of mapping class groups of infinite-type surfaces, now commonly known as big mapping class groups.…”

“…For infinite-type surfaces, Question 4.5 of the AIM Problem List on Surfaces of Infinite Type [1] (see also [5,Question 5.2]) asks: Question 1.1. Are big mapping class groups co-Hopfian?…”

“…While it is known that every automorphism of curve graph of an infinite-type surface is induced by a homeomorphism, [16,7], it is easy to see that this is no longer true if one drops the requirement that the map be surjective; see [18,5] for concrete examples. Our final result highlights the extent of this failure: Theorem 4.…”

Isospectral hyperbolic surfaces of infinite genus

“…Surfaces without boundary are topologically classified by their genus and the pair of topological spaces (Ends(S), Ends g (S)), where Ends(S) is the space of ends of the surface and Ends g (S) is the (closed) subspace of nonplanar ends, as shown by Kerékjártó [Ker23] and Richards [Ric63]. We refer to Aramayona and Vlamis' survey [AV20] for definitions and properties of these objects.…”

Curve graphs of surfaces with finite-invariance index 1

“…We review the classification of infinite type surfaces as well as notions related to the finite-invariance index, the Mann-Rafi partial order, self-similarity, and coarse boundedness of a group. For more comprehensive treatments of these topics, we refer the reader to [3], [13], [6], and [11].…”