The word metric of a group with respect to a generating set

Depends on

• Word length of a group element with respect to a generating set
• Word length is defined on every element and satisfies the subadditivity, inversion and vanishing laws
• Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric

Used by

• A group with the word metric of any generating set is a (1,1)-quasi-geodesic space Corollary
• Taking ℤ itself as a generating set gives a word metric of diameter one, not bilipschitz equivalent to the standard one Counterexample
• The quasi-isometry type of a finitely generated group Definition
• The Cayley graph of ℤ for the generating set {1} is a line and its word metric is |m-n| Example
• The Cayley graph of ℤⁿ for the standard basis is the integer lattice, and its word metric is the sum of coordinate differences Example
• The inclusion of ℤ in ℝ is a quasi-isometry that is neither surjective nor a bilipschitz equivalence Example
• The infinite dihedral group is quasi-isometric to ℤ, and to ℤ×ℤ/2 Example
• The subgroup 2ℤ×ℤ has index two in ℤ² and its inclusion is a quasi-isometry Example
• The word metrics of ℤ for {1} and for {2,3} differ at 1 and are bilipschitz equivalent Example
• FALSE: a nontrivial finitely generated group with a word metric is a geodesic metric space False statement
• FALSE: any two infinite finitely generated groups are quasi-isometric False statement
• FALSE: every word metric is invariant under right translation False statement
• A coarse Lipschitz map between word metric spaces of finitely generated groups is Lipschitz Proposition
• A subgroup of finite index in a finitely generated group is finitely generated, and its inclusion is a quasi-isometry Proposition
• Balls of a word metric are finite if and only if the generating set is finite Proposition
• Right translation by a fixed element displaces every point of a word metric space by exactly the word length of that element Proposition
• The quotient map by a finite normal subgroup is a quasi-isometry of word metric spaces Proposition
• The word metric is the largest left-invariant metric in which each generator and its inverse lie within distance one of the identity Proposition
• The identity map between the word metrics of two finite generating sets is a bilipschitz equivalence Theorem
• The word metric is a left-invariant metric and coincides with the path metric of the Cayley graph Theorem

Dependency tree · two levels

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Sources

• C. Drutu and M. Kapovich, Geometric Group Theory, Section 7.9 (standard reference, not scraped)
• C. Loh, Geometric Group Theory: An Introduction (2015 course version), Section 5.2 (standard reference, not scraped)