Michael Kapovich

The monodromy groups of Schwarzian equations on closed Riemann surfaces

Let θ : π1(R) → PSL(2, ℂ) be a homomorphism of the fundamental group of an oriented, closed surface R of genus exceeding one. We will establish the following theorem. THEOREM. Necessary and sufficient for θ to be the monodromy representation associated with a complex projective stucture on R, either unbranched or with a single branch point of order 2, is that θ(π1(R)) be nonelementary. A branch point is required if and only if the representation θ does not lift to SL(2, ℂ).

Hyperbolic groups with low-dimensional boundary

If a torsion-free hyperbolic group G has 1-dimensional boundary ∂∞G, then ∂∞G is a Menger curve or a Sierpinski carpet provided G does not split over a cyclic group. When ∂∞G is a Sierpinski carpet we show that G is a quasi-convex subgroup of a 3-dimensional hyperbolic Poincare duality group. We also construct a “topologically rigid” hyperbolic group G: any homeomorphism of ∂∞G is induced by an element of G.

On asymptotic cones and quasi-isometry classes of fundamental groups of 3-manifolds

We apply the concept of asymptotic cone to distinguish quasi-isometry classes of fundamental groups of 3-manifolds. We prove that the existence of a Seifert component in a Haken manifold is a quasi-isometry invariant of its fundamental group.

A path model for geodesics in Euclidean buildings and its applications to representation theory

In this paper we give a combinatorial characterization of projections of geodesics in Euclidean buildings to Weyl chambers. We apply these results to the representation theory of complex reductive Lie groups and to spherical Hecke rings associated with split nonar- chimedean reductive Lie groups. Our main application is a generalization of the saturation theorem of Knutson and Tao for SLn to other complex semisimple Lie groups.

Universality theorems for configuration spaces of planar linkages

We prove realizability theorems for vector-valued polynomial mappings, real-algebraic sets and compact smooth manifolds by moduli spaces of planar linkages. We also establish a relation between universality theorems for moduli spaces of mechanical linkages and projective arrangements.

On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties

We prove that for any affine variety S defined overQ there exist Shephard and Artin groups G such that a Zariski open subset U of S is biregular isomorphic to a Zariski open subset of the character variety X(G, PO(3)) = Hom(G, PO(3))//PO(3). The subset U contains all real points of S. As an application we construct new examples of finitely-presented groups which are not fundamental groups of smooth complex algebraic varieties.

The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra

In this paper we apply our results on the geometry of polygons in infinitesimal symmetric spaces, symmetric spaces and buildings, [KLM1, KLM2], to four problems in algebraic group theory. Two of these problems are generalizations of the problems of finding the constraints on the eigenvalues (resp. singular values) of a sum (resp. product) when the eigenvalues (singular values) of each summand (factor) are fixed. The other two problems are related to the nonvanishing of the structure constants of the (spherical) Hecke and representation rings associated with a split reductive algebraic group over Q and its complex Langlands’ dual. We give a new proof of the “Saturation Conjecture” for GL(l) as a consequence of our solution of the corresponding “saturation problem” for the Hecke structure constants for all split reductive algebraic groups over Q.

Actions of discrete groups on nonpositively curved spaces

We are interested in properties of groups which admit sufficiently nice actions by isometries on spaces of nonpositive curvature. Let us call a group Hadamard if it admits discrete actions by non-parabolic isometries on Hadamard spaces. These are synthetic analogues of Hadamard manifolds, they are complete geodesic metric spaces which are nonpositively curved in the sense of distance comparison. Typical examples of Hadamard groups are subgroups of fundamental groups of closed Riemannian manifolds of nonpositive sectional curvature. Various algebraic and geometric properties of Hadamard groups are well-known, such as: Solvable subgroups are virtually abelian [GW, LY] and centralizers virtually split [E, BH]. If G is a finitely generated Hadamard group then the stable norm (translation number) 11911 := l i m n ~ ~ of a non-periodic element g c G is always non-zero. This excludes for instance Baumslag-Solitar subgroups. The class of semisimple (i.e. non-parabolic) actions on Hadamard spaces has better functorial properties than the subclass of cocompact actions. Note however that there are groups which admit semisimple, but no cocompact discrete actions, e.g. certain infinitely generated groups or finitely generated groups with infinite-dimensional cohomology. Examples of such groups can be found in direct products of free groups, thus they have semisimple actions on products of hyperbolic planes. The main goal of this note is to find new obstructions for the existence of semisimple actions. In Sect. 2, we derive general properties of Hadamard groups, in particular:

Kleinian groups in higher dimensions

This is a survey of higher-dimensional Kleinian groups, i.e., discrete isometry groups of the hyperbolic n-space ℍ n for n ≥ 4. Our main emphasis is on the topological and geometric aspects of higher-dimensional Kleinian groups and their contrast with the discrete groups of isometry of ℍ3.

Quasi-isometries and the de Rham decomposition

Abstract We study quasi-isometries Φ:∏Xi→∏Yj of product spaces and find conditions on the X i , Y j which guarantee that the product structure is preserved. The main result applies to universal covers of compact Riemannian manifolds with nonpositive sectional curvature. We introduce a quasi-isometry invariant notion of coarse rank for metric spaces which coincides with the geometric rank for universal covers of closed nonpositively curved manifolds. This shows that the geometric rank is a quasi-isometry invariant.