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We introduce the notion of a critical constant $c_{rt}$ for recurrence of random walks on $G$-spaces. For a subgroup $H$ of a finitely generated group $G$ the critical constant is an asymptotic invariant of the quotient $G$-space $G/H$. We show that for any infinite $G$-space $c_{rt}\ge 1/2$. We say that $G/H$ is very small if $c_{rt}\<1$. For a normal subgroup $H$ the quotient space $G/H$ is very small if and only if it is finite. However, we give examples of infinite very small $G$-spaces. We show also that critical constants for recurrence can be used...

We give a construction of homomorphisms from a group into the reals using random walks on the group. The construction is an alternative to an earlier construction that works in more general situations. Applications include an estimate on the drift of random walks on groups of subexponential growth admitting no nontrivial homomorphism to the integers and inequalities between the asymptotic drift and the asymptotic entropy. Some of the entropy estimates obtained have applications independent of the...

We show that there exists a finitely generated group of growth $\sim f$ for all functions $f:{ℝ}_{+}\to {ℝ}_{+}$ satisfying $f\left(2R\right)\le f{\left(R\right)}^2\le f\left({\eta }_{+}R\right)$ for all $R$ large enough and ${\eta }_{+}\approx 2.4675$ the positive root of $X^3-X^2-2X-4$. Set ${\alpha }_{-}=log2/log{\eta }_{+}\approx 0.7674$; then all functions that grow uniformly faster than $exp\left(R^{{\alpha }_{-}}\right)$ are realizable as the growth of a group. We also give a family of sum-contracting branched groups of growth $\sim exp\left(R^{\alpha }\right)$ for a dense set of $\alpha \in [{\alpha }_{-},1]$.