Centers of subgroups of big mapping class groups and the Tits alternative

Justin Lanier; Marissa Loving

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arXiv · 1904.10060

Centers of subgroups of big mapping class groups and the Tits alternative

Justin Lanier·Marissa Loving

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Abstract

In this note we show that many subgroups of mapping class groups of infinite-type surfaces without boundary have trivial centers, including all normal subgroups. Using similar techniques, we show that every nontrivial normal subgroup of a big mapping class group contains a nonabelian free group. In contrast, we show that no big mapping class group satisfies the strong Tits alternative enjoyed by finite-type mapping class groups. We also give examples of big mapping class groups that fail to satisfy even the classical Tits alternative and give a proof that every countable group appears as a subgroup of some big mapping class group.

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Justin Lanier, Marissa Loving. 2019-04-22. Centers of subgroups of big mapping class groups and the Tits alternative. https://arxiv.org/abs/1904.10060

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