The critical constant for recurrence, $c_{rt}$, is an invariant of the quotient space $H/G$ of a finitely generated group. The constant is determined by the largest moment a probability measure on $G$ can have without the induced random walk on $H/G$ being recurrent. We present a description of which subgroups of groups of polynomial volume growth are recurrent. Using this we show that for such recurrent subgroups $c_{rt}$ corresponds to the relative growth rate of $H$ in $G$, and in particular $c_{rt}$ is either $0$, $1$ or $2$.
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机译：递归的临界常数$ c_ {rt} $是有限生成组的商空间$ H / G $的不变式。该常数由对$ G $的概率度量可能具有的最大矩确定，而不会导致对$ H / G $的诱导随机游走重复出现。我们介绍了多项式体积增长组的哪些子组是经常出现的。使用该函数，我们表明对于此类循环子组$ c_ {rt} $对应于$ H $在$ G $中的相对增长率，尤其是$ c_ {rt} $为$ 0 $，$ 1 $或$ 2 $。
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