Alphabeta MathHow statement and proof provenance work
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Word length is defined on every element and satisfies the subadditivity, inversion and vanishing laws
Statement
Word length is defined on every element and satisfies the subadditivity, inversion and vanishing laws.
Facts & Assumptions
Given: The hypotheses of the Statement.
The word length is the least such that is a product of elements of (Word length of a group element with respect to a generating set).
Every nonempty subset has a least element: there is with for all . (The well-ordering principle).
Proof
If generates, the set of lengths of expressions of is a nonempty subset of the natural numbers, so it has a least element.
Concatenating expressions gives subadditivity, and reversing an expression while inverting each letter gives equality of the lengths of and .
The empty expression has length zero and represents only the identity, so word length vanishes exactly there.
Depends on
- Word length of a group element with respect to a generating set
- The subgroup $\langle S \rangle$ generated by a subset, the cyclic subgroup $\langle g \rangle$, and cyclic groups
- The well-ordering principle
- In a group $e^{-1} = e$, $(g^{-1})^{-1} = g$ and $(gh)^{-1} = h^{-1}g^{-1}$, the order of the last product being essential
Used by
- With respect to a free basis, the word length of an element is the length of its reduced word Corollary
- The word metric of a group with respect to a generating set Definition
- A subgroup of finite index in a finitely generated group is finitely generated, and its inclusion is a quasi-isometry 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
21 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
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)