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Abstract We give a new proof of a theorem of D. Calegari that says that the Cayley graph of a surface group with respect to any generating set lying in finitely many mapping class group orbits has infinite diameter. This applies, for instance, to the generating set consisting of all simple closed curves.
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Graphs of curves for surfaces with finite-invariance index \(1\)
In this note we make progress toward a conjecture of Durham–Fanoni–Vlamis, showing that every infinite-type surface with finite-invariance index \(1\) and no nondisplaceable compact subsurfaces fails to have a good graph of curves, that is, a connected graph where vertices represent homotopy classes of essential simple closed curves and with a natural mapping class group action having infinite diameter orbits. Our arguments use tools developed by Mann–Rafi in their study of the coarse geometry of big mapping class groups.
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- PAR ID:
- 10479441
- Publisher / Repository:
- Croatian Mathematical Society
- Date Published:
- Journal Name:
- Glasnik Matematicki
- Volume:
- 57
- Issue:
- 1
- ISSN:
- 0017-095X
- Page Range / eLocation ID:
- 119 to 128
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3336/gm.57.1.08
