We exhibit a closed aspherical 5-manifold of nonpositive curvature that fibers over a circle whose fundamental group is hyperbolic relative to abelian subgroups such that the fiber is a closed aspherical 4-manifold whose fundamental group is not hyperbolic relative to abelian subgroups.
We define a notion of semi-conjugacy between orientation-preserving actions of a group on the circle, which for fixed point free actions coincides with a classical definition of Ghys. We then show that two circle actions are semi-conjugate if and only if they have the same bounded Euler class. This settles some existing confusion present in the literature.
We study the convergence of volume-normalized Betti numbers in Benjamini-Schramm convergent sequences of non-positively curved manifolds with finite volume. In particular, we show that if $X$ is an irreducible symmetric space of noncompact type, $X eq mathbb H^3$, and $(M_n)$ is any Benjamini-Schramm convergent sequence of finite volume $X$-manifolds, then the normalized Betti numbers $b_k(M_n)/vol(M_n)$ converge for all $k$. As a corollary, if $X$ has higher rank and $(M_n)$ is any sequence of distinct, finite volume $X$-manifolds, the normalized Betti numbers of $M_n$ converge to the $L^2$ Betti numbers of $X$. This extends our earlier work with Nikolov, Raimbault and Samet, where we proved the same convergence result for uniformly thick sequences of compact $X$-manifolds.
In this article, we consider a closed rank one $C^infty$ Riemannian manifold $M$ of nonpositive curvature and its universal cover $X$. Let $b_t(x)$ be the Riemannian volume of the ball of radius $t>0$ around $xin X$, and $h$ the topological entropy of the geodesic flow. We obtain the following Margulis-type asymptotic estimates [lim_{tto infty}b_t(x)/frac{e^{ht}}{h}=c(x)] for some continuous function $c: Xto mathbb{R}$. We prove that the Margulis function $c(x)$ is in fact $C^1$. If $M$ is a surface of nonpositive curvature without flat strips, we show that $c(x)$ is constant if and only if $M$ has constant negative curvature.
This is an expository proof that, if $M$ is a compact $n$-manifold with no boundary, then the set of holonomies of strictly-convex real-projective structures on $M$ is a subset of $operatorname{Hom}(pi_1M,operatorname{PGL}(n+1,mathbb RR))$ that is both open and closed.