Pattern preserving quasi-isometries in lamplighter groups and other related groups

2024/03/06 by Tullia Dymarz, Dymarz, Tullia, Beibei Liu +5
Mathematics · #20F65 #FOS: Mathematics #Group Theory (math.GR) #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2403.03928

openalex publication_date 2024/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we explore the interplay between aspects of the geometry and algebra of three families of groups of the form B semidirect the integers Z, namely Lamplighter groups, solvable Baumslag-Solitar groups and lattices in SOL. In particular we examine what kind of maps are induced on B by quasi-isometries that coarsely permute cosets of the Z subgroup. By the results of Schwartz(1996) and Taback(2000) in the lattice in SOL and solvable Baumslag-Solitar cases respectively such quasi-isometries induce affine maps of B. We show that this is no longer true in the lamplighter case but the induced maps do share some features with affine maps.

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