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arXiv · 0811.1607

On $k$-free-like groups

A. Yu. Olshanskii·M. V. Sapir
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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 $n\to \infty$. 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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A. Yu. Olshanskii, M. V. Sapir. 2008-11-12. On $k$-free-like groups. https://arxiv.org/abs/0811.1607

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