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Building on the work of Mann and Rafi, we introduce an expanded definition of a telescoping2-manifold and proceed to study the homeomorphism group of a telescoping2-manifold. Our main result shows that it is strongly distorted. We then give a simple description of its commutator subgroup, which is index one, two, or four depending on the topology of the manifold. Moreover, we show its commutator subgroup is uniformly perfect with commutator width at most two, and we give a family of uniform normal generators for its commutator subgroup. As a consequence of the latter result, we show that for an (orientable) telescoping2-manifold, every (orientation-preserving) homeomorphism can be expressed as a product of at most 17 conjugate involutions. Finally, we provide analogous statements for mapping class groups.
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This content will become publicly available on April 25, 2026
Orientation-preserving homeomorphisms of Euclidean space are commutators
We prove that every orientation-preserving homeomorphism of Euclidean space can be expressed as a commutator of two orientation-preserving homeomorphisms. We give an analogous result for annuli. In the annulus case, we also extend the result to the smooth category in the dimensions for which the associated sphere has a unique smooth structure. As a corollary, we establish that every orientation-preserving diffeomorphism of the real line is the commutator of two orientation-preserving diffeomorphisms.
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- Award ID(s):
- 2212922
- PAR ID:
- 10585273
- Publisher / Repository:
- American Mathematical Society (AMS)
- Date Published:
- Journal Name:
- Proceedings of the American Mathematical Society, Series B
- Volume:
- 12
- Issue:
- 4
- ISSN:
- 2330-1511
- Format(s):
- Medium: X Size: p. 38-47
- Size(s):
- p. 38-47
- Sponsoring Org:
- National Science Foundation
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