Krull dimensions of rings of holomorphic functions

We prove that the Krull dimension of the ring of holomorphic functions of a connected complex manifold is at least continuum if it is positive.Comment: 6 pages. An error pointed out by Pete Clark is corrected. The stronger statement about the Krull dimension at least continuum is prove

Homological dimension and critical exponent of Kleinian groups

We prove that the relative homological dimension of a Kleinian group G does not exceed 1 + the critical exponent of G. As an application of this result we show that for a geometrically finite Kleinian group G, if the topological dimension of the limit set of G equals its Hausdorff dimension, then the limit set is a round sphere.Comment: 38 page

A note on laminations with hyperbolic leaves

We prove that each solenoidal lamination with leaves isometric to the real-hyperbolic n-space and transitive homeomorphism group, is homeomorphic to the inverse limit of the system of finite covers of a compact hyperbolic n-manifold.Comment: 5 page

On sequences of finitely generated discrete groups

We consider sequences of finitely generated discrete subgroups Gamma_i=rho_i(Gamma) of a rank 1 Lie group G, where the representations rho_i are not necessarily faithful. We show that, for algebraically convergent sequences (Gamma_i), unless Gamma_i's are (eventually) elementary or contain normal finite subgroups of arbitrarily high order, their algebraic limit is a discrete nonelementary subgroup of G. In the case of divergent sequences (Gamma_i) we show that the limiting action on a real tree T satisfies certain semistability condition, which generalizes the notion of stability introduced by Rips. We then verify that the group Gamma splits as an amalgam or HNN extension of finitely generated groups, so that the edge group has an amenable image in the isometry group of T.Comment: 21 pages, 1 figur

Triangle inequalities in path metric spaces

We study side-lengths of triangles in path metric spaces. We prove that unless such a space X is bounded, or quasi-isometric to line or half-line, every triple of real numbers satisfying the strict triangle inequalities, is realized by the side-lengths of a triangle in X. We construct an example of a complete path metric space quasi-isometric to the Euclidean plane, for which every degenerate triangle has one side which is shorter than a certain uniform constant.Comment: 21 pages, 6 figure

RAAGs in Ham

We prove that every RAAG (a Right-Angled Artin Group) embeds in the group of Hamiltonian symplectomorphisms of the 2-sphere.Comment: 23 pages, 2 figure

Energy of harmonic functions and Gromov's proof of Stallings' theorem

We provide the details for Gromov's proof of Stallings' theorem on groups with infinitely many ends using harmonic functions. The main technical result of the paper is a compactness theorem for a certain family of harmonic functions.Comment: 4 figure

Arithmetic aspects of self-similar groups

We prove that an irreducible lattice in a semisimple algebraic group is virtually isomorphic to an arithmetic lattice if and only if it admits a faithful self-similar action on a rooted tree of finite valency.Comment: 6 page

Periods of abelian differentials and dynamics

Given a closed oriented surface S we describe those cohomology classes which appear as the period characters of abelian differentials for some choice of complex structure on S consistent with the orientation. The proof is based upon Ratner's solution of Raghunathan's conjecture.Comment: A revision of my 2000 preprin

A note on complex-hyperbolic Kleinian groups

Let $\Gamma$ be a discrete group of isometries acting on the complex hyperbolic $n$-space $\mathbb{H}^n_\mathbb{C}$. In this note, we prove that if $\Gamma$ is convex-cocompact, torsion-free, and the critical exponent $\delta(\Gamma)$ is strictly lesser than $2$, then the complex manifold $\mathbb{H}^n_\mathbb{C}/\Gamma$ is Stein. We also discuss several related conjectures.Comment: Minor revision. To appear in Arnold Math.