arXiv subjects
Publications and source records attributed to Michael Kapovich.
We prove that for every finitely-presented group G there exists a 2-dimensional irreducible complex-projective variety W with the fundamental group G, so that all singularities of W are normal crossings and Whitney umbrellas.
We prove that every RAAG (a Right-Angled Artin Group) embeds in the group of Hamiltonian symplectomorphisms of the 2-sphere.
We study fundamental groups of projective varieties with normal crossing singularities and of germs of complex singularities. We prove that for every finitely-presented group G there is a complex projective surface S with simple normal crossing singularities only, so that the fundamental group of S is isomorphic to G. We use this to construct 3-dimensional isolated complex singularities so that the fundamental group of the link is isomorphic to G. Lastly, we prove that a finitely-presented group G is Q-superperfect (has vanishing rational homology in dimensions 1 and 2) if and only if G is isomorphic to the fundamental group of the link of a rational 6-dimensional complex singularity.
We introduce the notion of ideal triangle in the Bruhat-Tits building associated to a split group -- it is analogous to the usual notion of triangle, but one vertex is "at infinity" in a certain direction. We prove that the algebraic variety of based ideal triangles with prescribed side-lengths is naturally isomorphic to a suitable variety of genuine triangles. From theorems pertaining to genuine triangles, we deduce saturation theorems related to branching to Levi subgroups and to the constant term homomorphisms.
The goal of the paper is to introduce a version of Schubert calculus for each dihedral reflection group W. That is, to each "sufficiently rich'' spherical building Y of type W we associate a certain cohomology theory and verify that, first, it depends only on W (i.e., all such buildings are "homotopy equivalent'') and second, the cohomology ring is the associated graded of the coinvariant algebra of W under certain filtration. We also construct the dual homology "pre-ring'' of Y. The convex "stability'' cones defined via these (co)homology theories of Y are then shown to solve the problem of classifying weighted semistable m-tuples on Y in the sense of Kapovich, Leeb and Millson equivalently, they are cut out by the generalized triangle inequalities for thick Euclidean buildings with the Tits boundary Y. Quite remarkably, the cohomology ring is obtained from a certain universal algebra A by a kind of "crystal limit'' that has been previously introduced by Belkale-Kumar for the cohomology of flag varieties and Grassmannians. Another degeneration of A leads to the homology theory of Y.
We prove, under the assumption of the virtual fibration conjecture for arithmetic hyperbolic 3-manifolds, that all arithmetic lattices in O(n,1), n> 4, and different from 7, are non-coherent. We also establish noncoherence of uniform arithmetic lattices of the simplest type in SU(n,1), n> 1, and of uniform lattices in SU(2,1) which have infinite abelianization.
We prove noncoherence of certain families of lattices in the isometry group of the hyperbolic n-space for n greater than 3. For instance, every nonuniform arithmetic lattice in SO(n,1) is noncoherent, provided that n is at least 6.
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.
We construct rank 2 thick nondiscrete affine buildings associated with an arbitrary finite dihedral group.
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 semisimple Lie groups and to spherical Hecke rings associated with nonarchimedean reductive Lie groups. Our main application is a generalization of the saturation theorem of Knutson and Tao for SL(n) to other complex semisimple Lie 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.
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.
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.
This is a survey of higher-dimensional Kleinian groups, i.e., discrete isometry groups of the hyperbolic n-space for n greater than 3. 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 the hyperbolic 3-space.
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.
We explicitly calculate the triangle inequalities for the group PSO(8). Therefore we explicitly solve the eigenvalues of sum problem for this group (equivalently describing the side-lengths of geodesic triangles in the corresponding symmetric space for the Weyl chamber-valued metric). We then apply some computer programs to verify two basic questions/conjectures. First, we verify that the above system of inequalities is irredundant. Then, we verify the ``saturation conjecture'' for the decomposition of tensor products of finite-dimensional irreducible representations of Spin(8). Namely, we show that for any triple of dominant weights a, b, c such that a+b+c is in the root lattice, and any positive integer N, the tensor product of the irreducible representations V(a) and V(b) contains V(c) if and only if the tensor product of V(Na) and V(Nb) contains V(Nc).
We consider Gromov-Thurston examples of negatively curved n-manifolds which do not admit metrics of constant sectional curvature. We show that for each n some of the Gromov-Thurston manifolds admit strictly convex real-projective structures.
As in a symmetric space of noncompact type, one can associate to an oriented geodesic segment in a Euclidean building a vector valued length in the Euclidean Weyl chamber; in addition to the metric length it contains information on the direction of the segment. We study in this paper restrictions on the vector valued side lengths of polygons in (thick) Euclidean buildings. The main result is that the set of possible side lengths of n-gons depends only on the associated spherical Coxeter complex. We show moreover that it coincides with the space of vector valued weights of semistable weighted configurations on the Tits boundary. The side lengths of polygons in symmetric spaces of noncompact type are studied in the related paper [KLM1]. Applications of the geometric results in both papers to algebraic group theory are given in [KLM3].