On a lemma of Milnor and Schwarz, apr\`es Rosendal

TL;DR

This paper revisits the Milnor--Schwarz lemma within Rosendal's framework, applying it to non-Archimedean groups and mapping class groups, and clarifying their quasi-isometry classifications.

Contribution

It provides a succinct treatment of the Milnor--Schwarz lemma in a topological group context and extends applications to non-Archimedean and mapping class groups.

Findings

Sharpened results on actions of big mapping class groups on hyperbolic graphs

Clarified quasi-isometry classification of certain mapping class groups

Extended the theory to locally finite infinite graphs and Stone space homeomorphisms

Abstract

Perhaps the fundamental theorem of geometric group theory, the Milnor--Schwarz lemma gives conditions under which the orbit map relating the geometry of a geodesic metric space and the word metric on a group acting isometrically on the space is a quasi-isometry. Pioneering work of Rosendal makes these and other techniques of geometric group theory applicable to an arbitrary (topological) group. We give a succinct treatment of the Milnor--Schwarz lemma, setting it within this context. We derive some applications of this theory to non-Archimedean groups, which have plentiful continuous actions on graphs. In particular, we sharpen results of BarNatan and Verberne on actions of "big" mapping class groups on hyperbolic graphs and clarify a project begun by Mann and Rafi to classify these mapping class groups up to quasi-isometry, noting some extensions to the theory of mapping class groups of locally finite infinite graphs and homeomorphism groups of Stone spaces.