Previously, we discussed that any big mapping class group is homeomorphic to the automorphism group of the curve graph. We use this today to prove that any big mapping class group is neither locally compact nor compactly generated. Further, we show that any big mapping class group is homeomorphic to the space of all irrational numbers. Finally, we will discuss the pure mapping class group and its topological generating sets.



References:


[AV20]     Javier Aramayona and Nicholas G. Vlamis. Big mapping class groups: an overview In In the tradition of Thurston-geometry and topology, pages 459-496. Springer, Cham, [2020] ⓒ2020. (arxiv version)

[YCV21]     Priyam Patel, Yassin Chandran, and Nicholas G. Vlamis. Infinite-type surfaces and mapping class groups: open problems, 2021.
Alexander method, Pure mapping class group, Dehn twist, Handle shift
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