Growth, isoperimetry and Liouville property for random walks on groups

In a joint work with Tianyi Zheng we show that the growth function of the first Grigorchuk group satisfies

lnlnvn/lnvn=a,

where a=log2/logx, x being a positive root of the polynomial x3−x2−2x−4. This is done by constructing measures with controlled tail decay and with non-trivial Poisson boundary. Such estimates are near optimal for all Grigorchuk groups Gw, and they can be far from being optimal for some other groups. It is essential that the above mentioned measures are infinitely supported. In contrast with subexponential growth groups, where all finitely supported measures have trivial boundary, general exponential growth groups are little understood. In a joint work with Josh Frisch we provide new criteria for solvable linear groups.