Orderability of big mapping class groups

Pravin Kumar; Apeksha Sanghi; Mahender Singh

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arXiv · 2407.14343

Orderability of big mapping class groups

Pravin Kumar·Apeksha Sanghi·Mahender Singh

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Abstract

We give an alternate proof of the left-orderability of the mapping class group of a connected oriented infinite-type surface with a non-empty boundary. Our main strategy involves the inductive construction of a countable stable Alexander system for the surface using a carefully chosen exhaustion by finite-type subsurfaces. In fact, we prove that a generalised ideal arc system for the surface also induces a left-ordering on the big mapping class group. We then prove that two generalised ideal arc systems determine the same left-ordering if and only if they are loosely isotopic. Finally, we prove that the topology on the big mapping class group is the same as the order topology induced by a left-ordering corresponding to an inductively constructed ideal arc system.

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Pravin Kumar, Apeksha Sanghi, Mahender Singh. 2024-07-19. Orderability of big mapping class groups. https://arxiv.org/abs/2407.14343

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