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Finiteness is a geometric property of finitely generated groups
Statement
Finiteness is a geometric property of finitely generated groups.
Facts & Assumptions
Given: The hypotheses of the Statement.
A quasi-isometry invariant is a map on finitely generated groups constant on quasi-isometry classes, and a property is geometric when its indicator is such an invariant (Quasi-isometry invariants and geometric properties of finitely generated groups).
A finitely generated group is finite if and only if it is quasi-isometric to a one-point space (A finitely generated group is finite if and only if it is quasi-isometric to a point).
A finitely generated group is quasi-isometric to a metric space when its word metric for some, equivalently every, finite generating set is (The quasi-isometry type of a finitely generated group).
Being quasi-isometric is a reflexive, symmetric and transitive relation on metric spaces (Being quasi-isometric is reflexive, symmetric and transitive).
Proof
Two quasi-isometric finitely generated groups are simultaneously quasi-isometric to a point, by transitivity of the relation.
Being quasi-isometric to a point characterises finiteness, so the indicator of finiteness is a quasi-isometry invariant and finiteness is a geometric property.
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Sources
- C. Loh, Geometric Group Theory: An Introduction (2015 course version), 264 pp. (standard reference, not scraped)
- C. Drutu and M. Kapovich, Geometric Group Theory (with an appendix by B. Nica), 837 pp. (standard reference, not scraped)