Behaviors of entropy on finitely generated groups

Details Behaviors of entropy on finitely generated groups Behaviors of entropy on finitely generated groups

2011-10-23

arxiv.org/abs/1110.5099v1

Mathematics

Group Theory

40 pages, 2 figures

Scientific paper

A variety of behaviors of entropy functions of random walks on finitely generated groups is presented, showing that for any $1/2 \leq \alpha \leq \beta \leq 1$, there is a group $\Gamma$ with measure $\mu$ equidistributed on a finite generating set such that $\liminf \frac{\log H_{\Gamma,\mu}(n)}{\log n}=\alpha$ and $\limsup \frac{\log H_{\G,\m}(n)}{\log n}=\beta$. The groups involved are finitely generated subgroups of the group of automorphisms of an extended rooted tree. The return probability and the drift of a simple random walk $Y_n$ on such groups are also evaluated, providing an exemple of group with return probability satisfying $\liminf \frac{\log |\log P(Y_n=_\Gamma 1)|}{\log n}=1/3$ and $\limsup \frac{\log |\log P(Y_n=_\G1)|}{\log n}=1$ and drift satisfying $\liminf \frac{\log E||Y_n||}{\log n}=1/2$ and $\limsup \frac{\log \E||Y_n||}{\log n}=1$.

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