Amenability of horocyclic products of percolation trees

TL;DR

This paper proves that horocyclic products of percolation subtrees of regular trees are almost surely amenable, and under symmetry conditions, they exhibit strong amenability with an explicit Foelner sequence.

Contribution

It establishes almost sure amenability and strong amenability with Foelner sequences for horocyclic products of percolation trees, extending understanding of their geometric properties.

Findings

Almost sure amenability of horocyclic products

Existence of a strong amenability with Foelner sequence under symmetry

Identification of conditions for strong amenability

Abstract

For horocyclic products of percolation subtrees of regular trees, we show almost sure amenability. Under a symmetry condition concerning the growth of the two percolation trees, we show the existence of an increasing Foelner sequence (which we call strong amenability).