Borel's stable range for the cohomology of arithmetic groups

In this note, we remark on the range in Borel's theorem on the stable cohomology of the arithmetic groups Sp(2n,Z) and SO(n,n;Z). This improves the range stated in Borel's original papers, an improvement that was known to Borel. Our main task is a technical computation involving the Weyl group action on roots and weights. This note originally appeared as the appendix to arXiv:1711.03139.Comment: Comments welcom

Geometric cycles and characteristic classes of manifold bundles

We produce new cohomology for non-uniform arithmetic lattices $\Gamma<SO(p,q)$ using a technique of Millson--Raghunathan. From this, we obtain new characteristic classes of manifold bundles with fiber a closed $4k$-dimensional manifold $M$ with indefinite intersection form of signature $(p,q)$. These classes are defined on a finite cover of $BDiff(M)$ and are shown to be nontrivial for $M=\#_g(S^{2k}\times S^{2k})$. In this case, the classes produced live in degree $g$ and are independent from the algebra generated by the stable (i.e. MMM) classes. We also give an application to bundles with fiber a K3 surface.Comment: New appendix by Manuel Krannich, which corrects and improves an argument in a previous draft. In addition, the original appendix on the stable range in Borel's theorem has been removed and will appear elsewhere (a preliminary draft is available on my website

Pontryagin classes of locally symmetric manifolds

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?Comment: 37 page

Cohomological obstructions to Nielsen realization

For a based manifold (M,*), the question of whether the surjection Diff(M,*) \rightarrow \pi_0 Diff(M,*) admits a section is an example of a Nielsen realization problem. This question is related to a question about flat connections on M-bundles and is meaningful for M of any dimension. In dimension 2, Bestvina-Church-Souto showed a section does not exist when M is closed and has genus g\ge 2. Their techniques are cohomological and certain aspects are specific to surfaces. We give new cohomological techniques to generalize their result to many locally symmetric manifolds. The main tools include Chern-Weil theory, Milnor-Wood inequalities, and Margulis superrigidity.Comment: 27 pages, final version, accepted for publication by the Journal of Topolog

Symmetries of exotic negatively curved manifolds

Let $N$ be a smooth manifold that is 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 $M$. Our main results are examples of $N$ that exhibit two extremes of behavior. 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 $N$. The construction of these examples incorporates a variety of techniques including smoothing theory and the Belolipetsky-Lubotzky method for constructing hyperbolic manifolds with a prescribed isometry group.Comment: 14 page

On the non-realizability of braid groups by diffeomorphisms

For every compact surface $S$ of finite type (possibly with boundary components but without punctures), we show that when $n$ is sufficiently large there is no lift $\sigma$ of the surface braid group $B_n(S)$ to $\operatorname{Diff}(S,n)$, the group of $C^1$ diffeomorphisms preserving $n$ marked points and restricting to the identity on the boundary. 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.Comment: This version incorporates a number of improvements as suggested by an anonymous referee. Of primary interest among this is the inclusion of a new proof of the Morita non-lifting theorem for $C^1$ diffeomorphisms for all $g\ge 2

On groups with S2S^2 Bowditch boundary

We prove that a relatively hyperbolic pair $(G,P)$ has Bowditch boundary a 2-sphere if and only if it is a 3-dimensional Poincare duality pair. We prove this by studying the relationship between the Bowditch and Dahmani boundaries of relatively hyperbolic groups.Comment: 19 pages. New version contains the converse to our main result (Corollary 3) and a section relating our work to the Wall and relative Cannon conjecture

Arithmeticity of groups Zn⋊Z\mathbb Z^n\rtimes\mathbb Z

We study when the group $\mathbb Z^n\rtimes_A\mathbb Z$ is arithmetic where $A\in GL_n(\mathbb Z)$ is hyperbolic and semisimple. We begin by giving a characterization of arithmeticity phrased in the language of algebraic tori, building on work of Grunewald-Platonov. We use this to prove several more concrete results that relate the arithmeticity of $\mathbb Z^n\rtimes_A\mathbb Z$ to the reducibility properties of the characteristic polynomial of $A$. Our tools include algebraic tori, representation theory of finite groups, Galois theory, and the inverse Galois problem.Comment: 22 pages. Comments welcom

Hyperbolic groups with boundary an n-dimensional Sierpinski space

For 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 nonempty boundary. Concerning the converse, we construct, for each n>3, examples of aspherical manifolds with boundary, whose fundamental group G is hyperbolic, but with visual boundary not homeomorphic to an (n-2)-dimensional Sierpinski space.Comment: 11 page

Arithmeticity of the monodromy of some Kodaira fibrations

A question of Griffiths-Schmid asks when the monodromy group of an algebraic family of complex varieties is arithmetic. We resolve this in the affirmative for the class of algebraic surfaces known as Atiyah-Kodaira manifolds, which have base and fibers equal to complete algebraic curves. Our methods are topological in nature and involve an analysis of the "geometric" monodromy, valued in the mapping class group of the fiber.Comment: 44 pages, 9 figure