Kathryn Mann

Homomorphisms between diffeomorphism groups

For r at least 3, p at least 2, we classify all actions of the groups Diff^r_c(R) and Diff^r_+(S1) by C^p -diffeomorphisms on the line and on the circle. This is the same as describing all nontrivial group homomorphisms between groups of compactly supported diffeomorphisms on 1- manifolds. We show that all such actions have an elementary form, which we call topologically diagonal. As an application, we answer a question of Ghys in the 1-manifold case: if M is any closed manifold, and Diff(M)_0 injects into the diffeomorphism group of a 1-manifold, must M be 1 dimensional? We show that the answer is yes, even under more general conditions. Several lemmas on subgroups of diffeomorphism groups are of independent interest, including results on commuting subgroups and flows.

Automatic continuity for homeomorphism groups and applications

Let M be a compact manifold, possibly with boundary. We show that the group of homeomorphisms of M has the automatic continuity property: any homomorphism from Homeo(M) to any separable group is necessarily continuous. This answers a question of C. Rosendal. If N is a submanifold of M, the group of homeomorphisms of M that preserve N also has this property. Various applications of automatic continuity are discussed, including applications to the topology and structure of groups of germs of homeomorphisms. In an appendix with Frederic Le Roux we also show, using related techniques, that the group of germs at a point of homeomorphisms of Rn is strongly uniformly simple.

A counterexample to the simple loop conjecture for PSL(2, ℝ)

In this note, we give an explicit counterexample to the simple loop conjecture for representations of surface groups into PSL.2;R/. Specifically, we use a construction of DeBlois and Kent to show that for any orientable surface with negative Euler characteristic and genus at least 1, there are uncountably many nonconjugate, noninjective homomorphisms of its fundamental group into PSL.2;R/ that kill no simple closed curve (nor any power of a simple closed curve). This result is not new — work of Louder and Calegari for representations of surface groups into SL.2;C/ applies to the PSL.2;R/ case, but our approach here is explicit and elementary.

Bounded orbits and global fixed points for groups acting on the plane

Let G be a group acting on the plane by orientation-preserving homeomorphisms. We show that if for some k>0 there is a ball of radius r > k/\sqrt{3} such that each point x in the ball satisfies |gx -hx| < k for all g, h in G, and the action of G satisfies a nonwandering hypothesis, then the action has a global fixed point. In particular, any group of measure-preserving orientation preserving homeomorphisms of the plane with uniformly bounded orbits has a global fixed point. The constant k/\sqrt{3} is sharp. We also show that a group acting on the plane with orbits bounded as above is left orderable.

Strong distortion in transformation groups

We show that the groups Diff0r(Rn) and Diffr(Rn) have the strong distortion property, whenever 0⩽r⩽∞,r≠n+1. This implies in particular that every element in these groups is distorted, a property with dynamical implications. The result also gives new examples of groups with Bergmans strong boundedness property as in Bergman (Bull. Lond. Math. Soc. 38 (2006) 429–440). With related techniques we show that, for M a closed manifold or homeomorphic to the interior of a compact manifold with boundary, the diffeomorphism groups Diff0r(M) satisfy a relative Higman embedding type property, introduced by Schreier. In the simplest case, this answers a problem asked by Schreier in the famous Scottish Book.