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

Tullia Dymarz; Beibei Liu; Nataša Macura; Rose Morris-Wright

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arXiv · 2403.03928

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

Tullia Dymarz·Beibei Liu·Nataša Macura·Rose Morris-Wright

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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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Tullia Dymarz, Beibei Liu, Nataša Macura, Rose Morris-Wright. 2024-03-06. Pattern preserving quasi-isometries in lamplighter groups and other related groups. https://arxiv.org/abs/2403.03928

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