On k-free-like groups

2008/11/11 by A. Yu. Olshanskii, Olshanskii, A. Yu., Mark Sapir +1
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.0811.1607

openalex publication_date 2008/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A k-free like group is a k-generated group G with a sequence of k-element generating sets Z_n such that the girth of G relative to Z_n is unbounded and the Cheeger constant of G relative to Z_n is bounded away from 0. By a recent result of Benjamini-Nachmias-Peres, this implies that the critical bond percolation probability of the Cayley graph of G relative to Z_n tends to 1/(2k-1) as n→ ∞. Answering a question of Benjamini, we construct many non-free groups that are k-free like for all sufficiently large k.

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