This page contains my research papers. Frequently the version on this page is more up-to-date than the version on the arXiv.

# Mapping class groups, arithmetic groups, characteristic classes

In the 1960s Atiyah and Kodaira constructed surface bundles over surfaces with many interesting properties. The topology of such a bundle is completely encoded in its monodromy representation (a homomorphism to a mapping class group), and it is a fundamental problem to understand precisely how the topology of the bundle is reflected in algebraic properties of the monodromy. The main result of this paper is that the Atiyah–Kodaira bundles have arithmetic monodromy groups. This addresses a question of Griffiths-Schmid. As an application, using work of L. Chen, one can show that Atiyah-Kodaira bundles fiber in exactly two ways. This is joint work with Nick Salter.

A basic problem in the study of fiber bundles is to compute the ring H*(BDiff(M)) of characteristic classes of bundles with fiber a smooth manifold M. When M is a surface, this problem has ties to algebraic topology, geometric group theory, and algebraic geometry. We have a good understanding of the cohomology in the *stable range, *but this accounts for a small percentage of the total cohomology, and little is known beyond that. In this paper we produce new classes (independent from the stable classes) for some special M that come from the unstable cohomology of arithmetic groups. Dually these nontrivial characteristic classes imply the existence of new topologically interesting manifold bundles. We also give an application to bundles with fiber a K3 surface.

This paper is about cohomology of mapping class groups of surfaces. 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 centralizer Mod(*S*)^*G* in Mod(*S*). There is a homomorphism R: Mod(*S*)^*G* --> Sp(2*g*,Z)^*G*. The main result of this paper is a computation of the induced map on cohomology in degree 2 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.

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

Mich. Math. J. 2018

# Aspherical manifolds

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 natural boundary -- the Bowditch boundary -- does not give a Z-set compactification of *G*. Nevertheless we show that if the Bowditch boundary of (*G,P*) is S^2, then (*G*,*P*) is a PD(3) pair.

**Groups with 2-sphere Bowditch boundary(with Genevieve Walsh)**

Submitted.

preprint

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.

**Hyperbolic groups with boundary an n-dimensional Sierpinski space(with Jean Lafont)**

J. Topol. Anal. to appear

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).

# Nielsen realization problems

**Symmetries of exotic negatively curved manifolds(with M. Bustamante)**

Let N be a smooth manifold that’s homeomorphic but not diffeomorphic to a closed hyperbolic manifold M. In this paper, we study the extent to which N admits as much symmetry as M. On the one hand, we find N with maximal symmetry, i.e. Isom(M) acts on N by isometries with respect to some negatively curved metric on N. For these examples, Isom(M) can be made arbitrarily large. On the other hand, we find N with little symmetry, i.e. no subgroup of Isom(M) of “small” index acts by diffeomorphisms of N.

This survey article discusses recent results and open questions on the broad theme of Nielsen realization problems with an emphasis on infinite subgroups of the mapping class group.

**Realization problems for diffeomorphism groups(with Kathryn Mann)**

accepted to Proc. Georgia Topol. Conf.

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.

**On the non-realizability of braid groups by diffeomorphisms(with Nick Salter)**

Bull. Lond. Math. Soc. 2016

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.