  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.
