2017/08/23 by John J. Harrison, Harrison, John J.
Computer Science · Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Geometric and Algebraic Topology #Probability (math.PR) #Topological and Geometric Data Analysis #math.PR
paper · pdf · doi:10.48550/arxiv.1708.06907
openalex publication_date 2017/08/23 · arxiv created 2017/08/24 · arxiv updated 2017/08/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We define matrix groups FGn(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 GLn (ℚ) for measures of finite first moment with respect to adelic length. We show that adelic length is a word metric estimate on FGn(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 FGn(P). We also investigate random walks in the case that P is a length one sequence. Conditions for pointwise convergence in ℝ or ℚp are given. When these conditions are satisfied, we give path estimates from boundary points, discuss boundary triviality, show that the resulting space is a μ-boundary and give cases where the μ-boundary is the \PF boundary, as conjectured by Kaimanovich.