Secret sharing on the Poisson-Furstenberg boundary I

TL;DR

This paper explores a surprising secret-sharing phenomenon in random walks on certain amenable groups, revealing non-trivial Poisson boundaries despite trivial marginals, expanding understanding of boundary behaviors in group theory.

Contribution

It demonstrates that secret-sharing phenomena occur in pairs of amenable groups with non-trivial ICC-factors, extending previous results to new group structures.

Findings

Secret-sharing occurs in pairs of amenable groups with ICC-factors.

Non-trivial Poisson boundary can coexist with trivial marginals.

The phenomenon is linked to specific group properties and measure behaviors.

Abstract

Recently, Vadim Kaimanovich presented a particular example of a measure on a product of two standard lamplighter groups such that the Poisson boundary of the induced random walk is non-trivial, but the boundary on the marginals is trivial. This was surprising since such behavior is not possible for measures of finite entropy. As we show in this paper, this secret-sharing phenomenon is possible precisely for pairs of amenable groups with non-trivial ICC-factors.