2025/11/05 by Robert Alonzo Lyman, Rylee Alanza Lyman, Lyman, Robert Alonzo · 1 voice · 1 citation
Mathematics · #Advanced Operator Algebra Research #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.GR #math.GT
paper · pdf · doi:10.48550/arxiv.2511.03887
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.