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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.03928 |
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| _version_ | 1866914705693474816 |
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| author | Dymarz, Tullia Liu, Beibei Macura, Nataša Morris-Wright, Rose |
| author_facet | Dymarz, Tullia Liu, Beibei Macura, Nataša Morris-Wright, Rose |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03928 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pattern preserving quasi-isometries in lamplighter groups and other related groups Dymarz, Tullia Liu, Beibei Macura, Nataša Morris-Wright, Rose Group Theory 20F65 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. |
| title | Pattern preserving quasi-isometries in lamplighter groups and other related groups |
| topic | Group Theory 20F65 |
| url | https://arxiv.org/abs/2403.03928 |