**Geometric cycles and manifold bundles**

In progress.

Much recent work has gone toward computing the ring H*( BDiff(M) ) of characteristic classes of manifold bundles. This is a difficult problem, but in many cases we have a good understanding of the cohomology in the "stable range". In this paper we produce classes outside the stable range coming from the unstable part of the cohomology of arithmetic groups.

**On groups whose Bowditch boundary is the 2-sphere(with Genevieve Walsh)**

In progress.

Bestvina-Mess showed that the duality properties of a group are encoded in any boundary that gives a Z-set compactification; in particular, a hyperbolic group with Gromov boundary S^n is a PD(n+1) group. For relatively hyperbolic pairs (*G*,*P*), the Bowditch boundary does not give a Z-set compactification. Nevertheless we show that if the Bowditch boundary of (G,P) is S^2, then (*G*,*P*) is a PD(3) pair.

## Characteristic classes of fiberwise branched surface bundles via arithmetic groups

Submitted.

Let *S* be a closed surface of genus *g*>1. The cohomology of its mapping class group Mod(*S*) and its finite-index subgroups plays an important role in many areas of mathematics. One source of cohomology classes is the symplectic representation*R*: Mod(*S*) --> Sp(2*g*,Z). A well-known computation shows that the image of

R*: H*( Sp(2g,Z) ;Q) --> H*( Mod(S) ;Q) is the algebra generated by the odd Miller-Morita-Mumford (MMM) classes in the stable range. In this paper we extend this example. Let *G* < Diff(*S*) be a finite group, and consider the subgroup Mod^*G*(*S*) of Mod(*S*) of mapping classes that can be realized by diffeomorphisms that commute with *G*. The image of Mod^*G*(*S*) under *R* lands in the centralizer Sp^*G*(2*g*,Z). The main result of this paper is a computation of *R**: H^2( Sp^*G*(2*g*,Z) ;Q) --> H^2( Mod^*G*(*S*) ;Q)

when *G* is a finite cyclic group. We apply this computation to compute Toledo invariants of surface group representations to SU(*p*,*q*) arising from the Atiyah-Kodaira construction, and we show that classes in the image of *R** gives equivariant cobordism invariants for surface bundles with fiberwise *G* action, following Church-Farb-Thibault.

## On the non-realizability of braid groups by diffeomorphisms

(with Nick Salter)

Bull. Lond. Math. Soc. 2016

For every compact surface we show that when *n* is sufficiently large there is no lift of the surface braid group Br(*n*,*S*) to Diff(*S*,*n*), the group of diffeomorphisms preserving *n* marked points and restricting to the identity on the boundary. This generalizes work of Bestvina-Church-Souto, but our main tool (the Thurston stability theorem) is different. Our methods are applied to give a new proof of Morita's non-lifting theorem in the best possible range. These techniques extend to the more general setting of spaces of codimension-2 embeddings, and we obtain corresponding results for spherical motion groups, including the string motion group.

## Hyperbolic groups with boundary an n-dimensional Sierpinski space

(with Jean Lafont)

J. Topol. Anal. to appear

The fundamental group of a closed aspherical manifold *M* is an example of a Poincare duality group. Whether or not all finitely presented Poincare duality groups arise in this fashion is an open problem that goes back to Wall. This paper addresses a relative version of this problem for a special class of groups. Fixing *n*>6, we show that if *G* is a torsion-free hyperbolic group whose visual boundary is an (*n*-2)-dimensional Sierpinski space, then *G* = pi_1(*W*) for some aspherical n-manifold *W* with boundary. The proof involves the total surgery obstruction and is based on work of Bartels-Lueck-Weinberger. Concerning the converse, we construct examples of aspherical manifolds with boundary whose fundamental group is hyperbolic but with visual boundary not homeomorphic to a Sierpinski space.

In this note we compute low degree rational Pontryagin classes for every closed locally symmetric manifold of noncompact type. In particular, we answer the question: Which locally symmetric M have at least one nonzero Pontryagin class? This computation was motivated by a Nielsen realization problem. Specifically, in my paper "Cohomological obstructions to Nielsen realization", I show that Pontryagin classes are obstructions to realizing the point-pushing subgroup for nonpositively curved manifolds (and in particular locally symmetric manifolds of noncompact type).

For a manifold *M* and a subgroup *G* < Mod(*M*) of the mapping class group, the question of whether *G* can be lifted to the diffeomorphism group Diff(*M*) is an example of a Nielsen realization problem. This question is related to an existence question for flat connections on fiber bundles with with monodromy *G*. In this paper we consider the case *M* is a locally symmetric manifold with a basepoint and *G* is the *point-pushing subgroup*. The simplest instance -- when *M* is a closed orientable surface of genus at least 2 -- was worked out by Bestvina-Church-Souto, who showed a lift does not exist. We give new cohomological techniques to generalize their result to higher dimensional locally symmetric manifolds. The main tools include Chern-Weil theory, Milnor-Wood inequalities, and Margulis superrigidity.