# On $k$-free-like groups - Mathematics > Group Theory

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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 awayfrom 0. By a recent result of Benjamini-Nachmias-Peres, this implies that thecritical 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 sufficientlylarge $k$.

Autor: A.Yu. Olshanskii, M. V. Sapir

Fuente: https://arxiv.org/