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.Comment: 68 pages 15 figure

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.Comment: 45 pages, 15 figures. See also http://www.math.utah.edu/~kapovich/eprints.htm

On representation varieties of 3-manifold groups

We prove universality theorems ("Murphy's Laws") for representation schemes of fundamental groups of closed 3-dimensional manifolds. We show that germs of SL(2,C)-representation schemes of such groups are essentially the same as germs of schemes of over rational numbers.Comment: 28 page

Matroids and Geometric Invariant Theory of torus actions on flag spaces

We apply a theorem of Gel'fand, Goresky, MacPherson, and Serganova about matroid polytopes to study semistability of partial flags relative to a T-linearized ample line bundle of a flag space F = SL(n)/P where T is a maximal torus in SL(n) and P is a parabolic subgroup containing T. We find that the semistable points are all detected by invariant sections of degree one regardless of the line bundle or linearization thereof, provided there exists at least one nonzero invariant section of degree one. In this case the degree one sections are sufficient to give a well defined map from the G.I.T. quotient F//T to projective space. Additionally, we show that the closure of any T-orbit in SL(n)/P is a projectively normal toric variety for any projective embedding of SL(n)/P.Comment: 14 page

The Relative Lie Algebra Cohomology of the Weil Representation of SO(n,1)

In Part 1 of this paper we construct a spectral sequence converging to the relative Lie algebra cohomology associated to the action of any subgroup $G$ of the symplectic group on the polynomial Fock model of the Weil representation, see Section 7. These relative Lie algebra cohomology groups are of interest because they map to the cohomology of suitable arithmetic quotients of the symmetric space $G/K$ of $G$. We apply this spectral sequence to the case $G = \mathrm{SO}_0(n,1)$ in Sections 8, 9, and 10 to compute the relative Lie algebra cohomology groups $H^{\bullet} \big(\mathfrak{so}(n,1), \mathrm{SO}(n); \mathcal{P}(V^k) \big)$. Here $V = \mathbb{R}^{n,1}$ is Minkowski space and $\mathcal{P}(V^k)$ is the subspace of $L^2(V^k)$ consisting of all products of polynomials with the Gaussian. In Part 2 of this paper we compute the cohomology groups $H^{\bullet}\big(\mathfrak{so}(n,1), \mathrm{SO}(n); L^2(V^k) \big)$ using spectral theory and representation theory. In Part 3 of this paper we compute the maps between the polynomial Fock and $L^2$ cohomology groups induced by the inclusions $\mathcal{P}(V^k) \subset L^2(V^k)$.Comment: 64 pages, 5 figure