Random walks on solvable matrix groups

TL;DR

This paper introduces matrix groups related to rational upper triangular matrices, explores their geometric properties, and analyzes the behavior of random walks on these groups, including boundary convergence and measure conditions.

Contribution

It constructs a computable word metric estimate for these groups, relates it to adelic length, and investigates boundary behavior of random walks, extending prior work on $GL_n(Q)$.

Findings

Finite first moment conditions are equivalent for adelic length and word length.

Conditions for convergence of random walks in $R$ and $Q_p$ are established.

Identifies when the boundary is trivial or coincides with the Poisson-Furstenberg boundary.

Abstract

We define matrix groups $FG_n(P)$ for each natural number $n$ and finite set of primes $P$, such that every rational-valued upper triangular matrix group is a (possibly distorted) subgroup. Brofferio and Schapira [Brofferio2011poisson], described the \PF boundary of $GL_n (\mathbb{Q})$ for measures of finite first moment with respect to adelic length. We show that adelic length is a word metric estimate on $FG_n(P)$ by constructing another, intermediate, word metric estimate which can be easily computed from the entries of any matrix in the group. In particular, finite first moment of a probability measure with respect to adelic length is an equivalent condition to requiring finite first moment with respect to word length in $FG_n(P)$. We also investigate random walks in the case that $P$ is a length one sequence. Conditions for pointwise convergence in $\mathbb{R}$ or $\mathbb{Q}_p$ are given. When these conditions are satisfied, we give path estimates from boundary points, discuss boundary triviality, show that the resulting space is a $\mu$-boundary and give cases where the $\mu$-boundary is the \PF boundary, as conjectured by Kaimanovich.