Note on the induced metric on power sets of groups
Tatsuhiko Yagasaki
TL;DR
This paper develops a general framework for extended, Hausdorff-type metrics on the power set ${\cal S}(G)$ of a group $G$ induced by a word-length norm and studies their algebraic and geometric properties. It introduces and analyzes asymmetric and multiplicative metrics, their symmetrizations, and the resulting quasi-isometric relationships between subspaces of ${\cal S}(G)$ under group homomorphisms, notably when $f$: $G\to H$ has kernel $K$ with ${\cal S}(K)$ metrically bounded. The work extends the theory to semidirect products, group actions, and $\ast$-sets, providing Lipschitz control and explicit lifts/sections that yield (quasi-)isometries between ${\cal S}$-spaces of the involved groups. The results offer a versatile metric framework for comparing subset structures across groups and for applications to coarse geometry and boundedness questions in diffeomorphism-related settings, unifying generation-based subset metrics with coarse geometric notions.
Abstract
In this note we clarify general properties of the Hausdorff-like metric on the power set ${\cal S}(G)$ of a group $G$ induced from word length norm and obtain some results on quasi-isometries between some subspaces of ${\cal S}(G)$ and ${\cal S}(H)$ for a group epimorphism $f : G \to H$, when ${\cal S}({\rm Ker}\,f)$ is metrically bounded.