BNSR-Invariants of Surface Houghton Groups
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915636804845568 |
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| author | Torgerson, Noah West, Jeremy |
| author_facet | Torgerson, Noah West, Jeremy |
| contents | The surface Houghton groups $\mathcal{H}_{n}$ are a family of groups generalizing Houghton groups $H_n$, which are constructed as asymptotically rigid mapping class groups. We give a complete computation of the BNSR-invariants $Σ^{m}(P\mathcal{H}_{n})$ of their intersection with the pure mapping class group. To do so, we prove that the associated Stein--Farley cube complex is CAT(0), and we adapt Zaremsky's method for computing the BNSR-invariants of the Houghton groups. As a consequence, we give a criterion for when subgroups of $H_n$ and $P\mathcal{H}_{n}$ having the same finiteness length as their parent group are finite index. We also discuss the failure of some of these groups to be co-Hopfian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04941 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | BNSR-Invariants of Surface Houghton Groups Torgerson, Noah West, Jeremy Geometric Topology Group Theory 20F65, 57K20, 57M07 The surface Houghton groups $\mathcal{H}_{n}$ are a family of groups generalizing Houghton groups $H_n$, which are constructed as asymptotically rigid mapping class groups. We give a complete computation of the BNSR-invariants $Σ^{m}(P\mathcal{H}_{n})$ of their intersection with the pure mapping class group. To do so, we prove that the associated Stein--Farley cube complex is CAT(0), and we adapt Zaremsky's method for computing the BNSR-invariants of the Houghton groups. As a consequence, we give a criterion for when subgroups of $H_n$ and $P\mathcal{H}_{n}$ having the same finiteness length as their parent group are finite index. We also discuss the failure of some of these groups to be co-Hopfian. |
| title | BNSR-Invariants of Surface Houghton Groups |
| topic | Geometric Topology Group Theory 20F65, 57K20, 57M07 |
| url | https://arxiv.org/abs/2403.04941 |