Peter A. Storm

Dense embeddings of surface groups

We discuss dense embeddings of surface groups and fully residually free groups in topological groups. We show that a compact topological group contains a nonabelian dense free group of finite rank if and only if it contains a dense surface group. Also, we obtain a characterization of those Lie groups which admit a dense faithfully embedded surface group. Similarly, we show that any connected semisimple Lie group contains a dense copy of any fully residually free group. 22E40; 20H10

Finiteness of arithmetic hyperbolic reflection groups

We prove that there are only finitely many conjugacy classes of arithmetic maximal hyperbolic reflection groups.

The barycenter method on singular spaces.

Compact convex cores with totally geodesic boundary are proven to uniquely minimize volume over all hyperbolic 3-manifolds in the same homotopy class. This solves a conjecture in Kleinian groups concerning acylindrical 3-manifolds. Closed hyperbolic manifolds are proven to uniquely minimize volume over all compact hyperbolic cone-manifolds in the same homotopy class with cone angles =2p. Closed hyperbolic manifolds are proven to minimize volume over all compact Alexandrov spaces with curvature bounded below by -1 in the same homotopy class. A version of the Besson?Courtois?Gallot theorem is proven for n-manifolds with boundary. The proofs extend the techniques of Besson?Courtois?Gallot.

Dynamics of the mapping class group action on the variety of PSL2C characters

We study the action of the mapping class group Mod.S/ on the boundary @Q of quasifuchsian space Q. Among other results, Mod.S/ is shown to be topologically transitive on the subset C @Q of manifolds without a conformally compact end. We also prove that any open subset of the character variety X. 1.S/; SL2 C/ intersecting @Q does not admit a nonconstant Mod.S/‐invariant meromorphic function. This is related to a question of Goldman. 57M50; 58D27

Moduli spaces of hyperbolic 3-manifolds and dynamics on character varieties

The space AH(M) of marked hyperbolic 3-manifold homotopy equivalent to a compact 3-manifold with boundary M sits inside the PSL2(C)-character variety X(M) of π1(M). We study the dynamics of the action of Out(π1(M)) on both AH(M) and X(M). The nature of the dynamics reflects the topology of M . The quotientAI(M) = AH(M)/Out(π1(M)) may naturally be thought of as the moduli space of unmarked hyperbolic 3-manifolds homotopy equivalent to M and its topology reflects the dynamics of the action.

The curious moduli spaces of unmarked Kleinian surface group

Fixing a closed hyperbolic surface

Hyperbolic convex cores and simplicial volume

S

Lipschitz minimality of Hopf fibrations and Hopf vector fields

, we define a moduli space

Local rigidity of hyperbolic manifolds with geodesic boundary

{\rm A}{\cal I}(S)

From the hyperbolic 24-cell to the cuboctahedron

of unmarked hyperbolic