Representations of polygons of finite groups
We construct discrete and faithful representations into the isometry group of a hyperbolic space of the fundamental groups of acute negatively curved even-sided polygons of finite groups.
math.GR↗
arXiv subjects
Michael Kapovich
Publications and source records attributed to Michael Kapovich.
Structure of the tensor product semigroup
We study structure of the semigroup Tens(G) consisting of triples of dominant weights (λ,μ,ν) of a complex reductive Lie group G such that the triple tensor product of the corresponding irreducible representations of G has a nonzero G-invariant vector. We prove two general structural results for Tens(G) and give an explicit computation of Tens(G) for G=Sp(4,C) and G=G_2.
math.RT↗
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 Cartan subspaces, symmetric spaces and buildings 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 computation of the structure constants of the 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(m) as a consequence of our solution of the corresponding ``saturation problem'' for the Hecke structure constants for all split reductive algebraic groups over Q.
math.RT↗
Conformally flat metrics on 4-manifolds
We prove that for each closed smooth spin 4-manifold M there exists a closed smooth 4-manifold N such that the connected sum M # N admits a conformally flat Riemannian metric.
math.DG↗
Van Kampen's embedding obstruction for discrete groups
We give a lower bound to the dimension of a contractible manifold on which a given group can act properly discontinuously. In particular, we show that the $n$-fold product of nonabelian free groups cannot act properly discontinuously on $\R^{2n-1}$.
math.GT↗
The symplectic geometry of polygons in hyperbolic 3-space
We study the symplectic geometry of the moduli space of closed n-gons with fixed side-lengths in hyperbolic 3-space. We prove that these moduli spaces have a symplectic structure coming from Poisson Lie theory. We construct completely integrable systems on these moduli spaces by bending n-gons along their diagonals. The results of this paper are the analogues of the results of the first two authors for n-gon linkages in Euclidean 3-space in JDG 44. We conclude by proving by a deformation argument that the moduli spaces of n-gon linkages in hyperbolic 3-space and Euclidean 3-space with the same set of side-lengths are symplectomorphic.
math.SG↗
The monodromy groups of Schwarzian equations on closed Riemann surfaces
Let θ:π_1(R) \to \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 θto 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(π_1(R)) be nonelementary. A branch point is required if and only if the representation θdoes not lift to \SL(2,\C).
math.CV↗
Quantization of bending deformations of polygons in Euclidean space, hypergeometric integrals and the Gassner representation
We quantize the bending deformations of n-gon linkages by linearizing the bending fields at a degenerate n-gon to get a representation of the Malcev Lie algebra of the pure braid group. This linearization yields a flat connection on the space of n distinct points on the complex line. We show that the monodromy is (essentially) the Gassner representation.
math.SG↗
Coarse Alexander duality and duality groups
We study discrete group actions on coarse Poincare duality spaces, e.g. acyclic simplicial complexes which admit free cocompact group actions by Poincare duality groups. When G is an (n-1) dimensional duality group and X is a coarse Poincare duality space of formal dimension n, then a free simplicial action of G on X determines a collection of ``peripheral'' subgroups F_1,...,F_k in G so that the group pair (G;F_1,...,F_k) is an n-dimensional Poincare duality pair. In particular, if G is a 2-dimensional 1-ended group of type FP_2, and X is a coarse PD(3) space, then G contains surface subgroups; if in addition X is simply connected, then we obtain a partial generalization of the Scott/Shalen compact core theorem to the setting of coarse PD(3) spaces.
math.GT↗
On normal subgroups in the fundamental groups of complex surfaces
We show that for each aspherical compact complex surface $X$ whose fundamental group $π$ fits into a short exact sequence $$ 1\to K \to π\to π_1(S) \to 1 $$ where $S$ is a compact hyperbolic Riemann surface and the group $K$ is finitely-presentable, there is a complex structure on $S$ and a nonsingular holomorphic fibration $f: X\to S$ which induces the above short exact sequence. In particular, the fundamental groups of compact complex-hyperbolic surfaces cannot fit into the above short exact sequence. As an application we give the first example of a non-coherent uniform lattice in $PU(2,1)$.
math.GT↗
Hyperbolic groups with 1-dimensional boundary
If a torsion-free hyperbolic group G has 1-dimensional boundary, then the boundary is a Menger curve or a Sierpinski carpet provided G does not split over a cyclic group. When the boundary of 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 the boundary of G is induced by an element of G.
math.GR↗
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.
math.AG↗
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 over Q 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 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.
dg-ga↗