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

2019/04/22 by Lanier, Justin, Loving, Marissa · 1 citation
#20F34 (Primary) 57M60 #37E30 (Secondary) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1904.10060

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.

Cited by

• Non-amenability of mapping class groups of infinite-type surfaces and graphs

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