Strongly bounded generation in transformation groups
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914306382102528 |
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| author | Vlamis, Nicholas G. |
| author_facet | Vlamis, Nicholas G. |
| contents | Word metrics on finitely generated groups have canonical quasi-isometry classes, making quasi-isometry invariants genuine group invariants. Rosendal generalized this phenomenon to topological groups through CB-generation, but in the general topological setting the resulting quasi-isometry invariants are not invariants of the underlying abstract group. Specializing to the discrete case yields what we call SB-generated groups, where the invariants are genuinely algebraic. We show that SB-generation arises naturally in transformation groups by identifying several broad families of examples: the identity component of homeomorphism groups of closed manifolds, certain big mapping class groups, and homeomorphism groups of compact well-ordered spaces with successor limit capacity. These results demonstrate that SB-generation provides a robust extension of finite generation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07541 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strongly bounded generation in transformation groups Vlamis, Nicholas G. Group Theory General Topology Geometric Topology Word metrics on finitely generated groups have canonical quasi-isometry classes, making quasi-isometry invariants genuine group invariants. Rosendal generalized this phenomenon to topological groups through CB-generation, but in the general topological setting the resulting quasi-isometry invariants are not invariants of the underlying abstract group. Specializing to the discrete case yields what we call SB-generated groups, where the invariants are genuinely algebraic. We show that SB-generation arises naturally in transformation groups by identifying several broad families of examples: the identity component of homeomorphism groups of closed manifolds, certain big mapping class groups, and homeomorphism groups of compact well-ordered spaces with successor limit capacity. These results demonstrate that SB-generation provides a robust extension of finite generation. |
| title | Strongly bounded generation in transformation groups |
| topic | Group Theory General Topology Geometric Topology |
| url | https://arxiv.org/abs/2510.07541 |