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We investigate the problem of when big mapping class groups are generated by involutions. Restricting our attention to the class of self-similar surfaces, which are surfaces with self-similar ends spaces, as defined by Mann and Rafi, and with 0 or infinite genus, we show that when the set of maximal ends is infinite, then the mapping class groups of these surfaces are generated by involutions, normally generated by a single involution, and uniformly perfect. In fact, we derive this statement as a corollary of the corresponding statement for the homeomorphism groups of these surfaces. On the other hand, among self-similar surfaces with one maximal end, we produce infinitely many examples in which their big mapping class groups are neither perfect nor generated by torsion elements. These groups also do not have the automatic continuity property.
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Automatic continuity of pure mapping class groups
We completely classify the orientable infinite-type surfaces S such that PMap(S), the pure mapping class group, has automatic continuity. This classification includes surfaces with noncompact boundary. In the case of surfaces with finitely many ends and no noncompact boundary components, we prove the mapping class group Map(S) does not have automatic continuity. We also completely classify the surfaces such that PMapc (S), the subgroup of the pure mapping class group composed of elements with representatives that can be approximated by compactly supported homeomorphisms, has automatic continuity. In some cases when PMapc (S) has automatic continuity, we show any homomorphism from PMapc (S) to a countable group is trivial.
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- Award ID(s):
- 1745583
- PAR ID:
- 10613999
- Publisher / Repository:
- State University of New York at Albany
- Date Published:
- Journal Name:
- The Womens business resource guide
- ISSN:
- 1076-9803
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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