We show that if M is a compact oriented surface of genus 0 and G is a subgroup of Sympωμ (M) that has an infinite normal solvable subgroup, then G is virtually abelian. In particular the centralizer of an infinite order f ∈ Sympωμ(M) is virtually abelian. Another immediate corollary is that if G is a solvable subgroup of Sympωμ(M) then G is virtually abelian. We also prove a special case of the Tits Alternative for subgroups of Sympωμ(M).
- Surface diffeomorphism groups
ASJC Scopus subject areas
- Algebra and Number Theory
- Applied Mathematics