Density of crystalline points on unitary Shimura varieties

We prove that crystalline points are dense in the spectrum of the completed Hecke algebra on unitary Shimura varieties.Comment: 11 pages, submitte

Vanishing resonance and representations of Lie algebras

We explore a relationship between the classical representation theory of a complex, semisimple Lie algebra \g and the resonance varieties R(V,K)\subset V^* attached to irreducible \g-modules V and submodules K\subset V\wedge V. In the process, we give a precise roots-and-weights criterion insuring the vanishing of these varieties, or, equivalently, the finiteness of certain modules W(V,K) over the symmetric algebra on V. In the case when \g=sl_2(C), our approach sheds new light on the modules studied by Weyman and Eisenbud in the context of Green's conjecture on free resolutions of canonical curves. In the case when \g=sl_n(C) or sp_{2g}(C), our approach yields a unified proof of two vanishing results for the resonance varieties of the (outer) Torelli groups of surface groups, results which arose in recent work by Dimca, Hain, and the authors on homological finiteness in the Johnson filtration of mapping class groups and automorphism groups of free groups.Comment: 17 pages; Corollary 1.3 stated in stronger form, with a shorter proo

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

A Jorgensen-Thurston theorem for homomorphisms

In this note, we provide a description of the structure of homomorphisms from a finitely generated group to any torsion-free (3-dimensional) Kleinian group with uniformly bounded finite covolume. This is analogous to the Jorgensen-Thurston Theorem in hyperbolic geometry.Comment: 16 pages, 4 figure

Fundamental Groups of Character Varieties: Surfaces and Tori

We compute the fundamental group of moduli spaces of Lie group valued representations of surface and torus groups.Comment: v2: 12 pages, minor edits, accepted for publication at Mathematische Zeitschrif

Support varieties for transporter category algebras

Let G be a finite group. Over any finite G-poset P we may define a transporter category as the corresponding Grothendieck construction. The classifying space of the transporter category is the Borel construction on the G-space BP, while the k-category algebra of the transporter category is the (Gorenstein) skew group algebra on the G-incidence algebra kP. We introduce a support variety theory for the category algebras of transporter categories. It extends Carlson's support variety theory on group cohomology rings to equivariant cohomology rings. In the mean time it provides a class of (usually non selfinjective) algebras to which Snashall-Solberg's (Hochschild) support variety theory applies. Various properties will be developed. Particularly we establish a Quillen stratification for modules.Comment: 22 pages. Removed some small errors. Added a Lemma 2.3.2 and 2 new references on Gorenstein skew group algebra