Geometries for Groups

Abstract

We prove a "continuous converse" to the Milnor-Schwarz lemma. Given any topological group $G$ and any left-invariant, large scale geodesic pseudometric $d$ giving an identity neighborhood finite diameter (for example, the word metric with respect to an open generating set), we construct a geodesic metric space $X$ equipped with a continuous, isometric, cobounded action of $G$ such that $(G,d)$ and $X$ are quasi-isometric via any orbit map.

arXiv commentv2: almost entirely rewritten with a new result and new title, 8 pages, prepared in Typst (dear arXiv, please accept Typst source!)