On a lemma of Milnor and Schwarz, après Rosendal
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912689999052800 |
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| author | Lyman, Robert Alonzo |
| author_facet | Lyman, Robert Alonzo |
| contents | Perhaps the fundamental theorem of geometric group theory, the Milnor--Schwarz lemma gives conditions under which the orbit map relating the geometry of a geodesic metric space and the word metric on a group acting isometrically on the space is a quasi-isometry.
Pioneering work of Rosendal makes these and other techniques of geometric group theory applicable to an arbitrary (topological) group. We give a succinct treatment of the Milnor--Schwarz lemma, setting it within this context. We derive some applications of this theory to non-Archimedean groups, which have plentiful continuous actions on graphs. In particular, we sharpen results of BarNatan and Verberne on actions of "big" mapping class groups on hyperbolic graphs and clarify a project begun by Mann and Rafi to classify these mapping class groups up to quasi-isometry, noting some extensions to the theory of mapping class groups of locally finite infinite graphs and homeomorphism groups of Stone spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03887 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a lemma of Milnor and Schwarz, après Rosendal Lyman, Robert Alonzo Group Theory Geometric Topology 20F65 Perhaps the fundamental theorem of geometric group theory, the Milnor--Schwarz lemma gives conditions under which the orbit map relating the geometry of a geodesic metric space and the word metric on a group acting isometrically on the space is a quasi-isometry. Pioneering work of Rosendal makes these and other techniques of geometric group theory applicable to an arbitrary (topological) group. We give a succinct treatment of the Milnor--Schwarz lemma, setting it within this context. We derive some applications of this theory to non-Archimedean groups, which have plentiful continuous actions on graphs. In particular, we sharpen results of BarNatan and Verberne on actions of "big" mapping class groups on hyperbolic graphs and clarify a project begun by Mann and Rafi to classify these mapping class groups up to quasi-isometry, noting some extensions to the theory of mapping class groups of locally finite infinite graphs and homeomorphism groups of Stone spaces. |
| title | On a lemma of Milnor and Schwarz, après Rosendal |
| topic | Group Theory Geometric Topology 20F65 |
| url | https://arxiv.org/abs/2511.03887 |