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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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Generating the homology of covers of surfaces
Abstract Putman and Wieland conjectured that if is a finite branched cover between closed oriented surfaces of sufficiently high genus, then the orbits of all nonzero elements of under the action of lifts to of mapping classes on are infinite. We prove that this holds if is generated by the homology classes of lifts of simple closed curves on . We also prove that the subspace of spanned by such lifts is a symplectic subspace. Finally, simple closed curves lie on subsurfaces homeomorphic to 2‐holed spheres, and we prove that is generated by the homology classes of lifts of loops on lying on subsurfaces homeomorphic to 3‐holed spheres.
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- PAR ID:
- 10504577
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Bulletin of the London Mathematical Society
- Volume:
- 56
- Issue:
- 5
- ISSN:
- 0024-6093
- Format(s):
- Medium: X Size: p. 1768-1787
- Size(s):
- p. 1768-1787
- Sponsoring Org:
- National Science Foundation
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