Glasnik Matematicki, Vol. 55, No. 1 (2020), 85-91.

CENTERS OF SUBGROUPS OF BIG MAPPING CLASS GROUPS AND THE TITS ALTERNATIVE

Justin Lanier and Marissa Loving

School of Mathematics, Georgia Institute of Technology, 686 Cherry St., Atlanta, GA 30332, USA
e-mail: jlanier8@gatech.edu

Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, USA
e-mail: mloving2@illinois.edu

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; consequently, these examples are not linear.

2010 Mathematics Subject Classification.   20F34, 57M60, 37E30

Key words and phrases.   Mapping class groups, normal subgroups, free subgroups, Tits alternative, centers

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https://doi.org/10.3336/gm.55.1.07

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