Related papers: Homology growth, hyperbolization, and fibering

In Arakelov theory a completion of an arithmetic surface is achieved by enlarging the group of divisors by formal linear combinations of the ``closed fibers at infinity''. Manin described the dual graph of any such closed fiber in terms of…

We study the intrinsic geometry of area minimizing (and also of almost minimizing) hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. For any such hypersurface we define and construct a so-called…

We show that (under mild assumptions) the generating function of log homology torsion of a knot exterior has a meromorphic continuation to the entire complex plane. As corollaries, this gives new proofs of (a) the Silver-Williams…

A hyperbolic group acts by homeomorphisms on its Gromov boundary. We use a dynamical coding of boundary points to show that such actions are topologically stable in the dynamical sense: any nearby action is semi-conjugate to (and an…

If a graph is in bridge position in a 3-manifold so that the graph complement is irreducible and boundary irreducible, we generalize a result of Bachman and Schleimer to prove that the complexity of a surface properly embedded in the…

The theory of complex hyperbolic discrete groups is still in its childhood but promises to grow into a rich subfield of geometry. In this paper I will discuss some recent progress that has been made on complex hyperbolic deformations of the…

We introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be thought of as a generalization of the peripheral structure of a relatively hyperbolic group, while the later one…

In this survey we discuss how geometric methods can be used to study topological properties of 3-manifolds such as their Heegaard genus or the rank of their fundamental group. On the other hand, we also discuss briefly some results relating…

The purpose of the present paper is to prove existence of super-exponentially many compact orientable hyperbolic arithmetic $n$-manifolds that are geometric boundaries of compact orientable hyperbolic $(n+1)$-manifolds, for any $n \geq 2$,…

This paper shows that every Gromov hyperbolic group can be described by a finite subdivision rule acting on the 3-sphere. This gives a boundary-like sequence of increasingly refined finite cell complexes which carry all quasi-isometry…

We exhibit the first examples of contact structures on $S^{2n-1}$ with $n\geq 4$ and on $S^3\times S^2$, all equipped with their standard smooth structures, for which every Reeb flow has positive topological entropy. As a new technical tool…

For each n greater than 7 we explicitly construct a sequence of Stein manifolds diffeomorphic to complex affine space of dimension n so that there is no algorithm to tell us in general whether a given such Stein manifold is symplectomorphic…

We prove that any diffeomorphism of a compact manifold can be C^1-approximated by a diffeomorphism which exhibits a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by a diffeomorphism which is partially…

We prove a finiteness theorem for subgroups of bounded rank in hyperbolic $3$-manifold groups. As a consequence, we show that every bounded rank covering tower of closed hyperbolic $3$-manifolds is a tower of finite covers associated to a…

Very recently, Kim and Lee presented an example of a non-Hopfian relatively hyperbolic group with a Hopfian peripheral subgroup, demonstrating a counterexample to Osin's well-known question (Problem 5.5). In this paper, we provide a general…

For each surface $S$ of genus $g>2$ we construct pairs of conjugate pseudo-Anosov maps, $\varphi_1$ and $\varphi_2$, and two non-equivalent covers $p_i: \tilde S \longrightarrow S$, $i=1,2$, so that the lift of $\varphi_1$ to $\tilde S$…

Let $M_0$ be a compact and orientable 3-manifold. After capping off spherical boundaries with balls and removing any torus boundaries, we prove that the resulting manifold $M$ contains handlebodies of arbitrary genus such that the closure…

Suppose n>2, let M,M' be n-dimensional connected complete finite-volume hyperbolic manifolds with non-empty geodesic boundary, and suppose that the fundamental group of M is quasi-isometric to the fundamental group of M' (with respect to…

We construct a number of topologically trivial but smoothly non-trivial families of embeddings of 3-manifolds in 4-manifolds. These include embeddings of homology spheres in $S^4$ that are not isotopic but have diffeomorphic complements,…

This paper contains some more results on the topology of a nondegenerate action of $\mathbb{R}^n$ on a compact connected $n$-manifold $M$ when the action is totally hyperbolic (i.e. its toric degree is zero). We study the…