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The quasi-isometry type of a finitely generated group
Definition
Let be a finitely generated group. Choose a finite generating set and form the word metric space of The word metric of a group with respect to a generating set.
By The identity map between the word metrics of two finite generating sets is a bilipschitz equivalence, the word metrics are bilipschitz equivalent; by Every isometry is a bilipschitz equivalence and every bilipschitz equivalence is a quasi-isometry, and two metrics on one set are Lipschitz equivalent exactly when the identity is a bilipschitz equivalence between them they are therefore quasi-isometric. Hence the quasi-isometry class of is independent of the chosen finite generating set .
This well-defined quasi-isometry class is the quasi-isometry type of .
Depends on
- Finitely generated groups
- The word metric of a group with respect to a generating set
- The identity map between the word metrics of two finite generating sets is a bilipschitz equivalence
- Every isometry is a bilipschitz equivalence and every bilipschitz equivalence is a quasi-isometry, and two metrics on one set are Lipschitz equivalent exactly when the identity is a bilipschitz equivalence between them
Used by
- Quasi-isometry invariants and geometric properties of finitely generated groups Definition
- The infinite dihedral group is quasi-isometric to ℤ, and to ℤ×ℤ/2 Example
- FALSE: any two infinite finitely generated groups are quasi-isometric False statement
- A finitely generated group is finite if and only if it is quasi-isometric to a point Proposition
- Finiteness is a geometric property of finitely generated groups Proposition
Dependency tree · two levels
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Sources
- C. Loh, Geometric Group Theory: An Introduction (2015 course version), Section 5.2 (standard reference, not scraped)