Related papers: Dynamics on nilpotent character varieties

Let $\Gamma$ be a finitely generated nilpotent group and let G be a complex reductive algebraic group. The representation variety $\mathrm{Hom}(\Gamma,G)$ and the character variety $\mathrm{Hom}(\Gamma,G)//G$ each carry a natural topology,…

Let R be the connected component of the identity of the variety of representations of a finitely generated nilpotent group N into a connected reductive complex affine algebraic group G. We determine the mixed Hodge structure on the…

When $G$ is a connected compact Lie group, and $\pi$ is a closed surface group, then $Hom(\pi,G)$ contains an open dense $Out(\pi)$-invariant subset which is a smooth symplectic manifold. This symplectic structure is $Out(\pi)$-invariant…

Let G be a connected complex reductive affine algebraic group, and let K be a maximal compact subgroup. Let X be a compact connected K\"ahler manifold whose fundamental group Gamma is virtually nilpotent. We prove that the character variety…

We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of GL_n(C) on the variety of x-nilpotent complex matrices. We obtain a criterion as to whether the action admits a finite number of orbits and specify a…

Let $\Gamma$ be a hyperbolic group and G be the isometry group of a Gromov-hyperbolic, properand geodesic metric space. We study the action of the outer automorphism group Out($\Gamma$) onthe set X($\Gamma$,G) of conjugacy classes of…

Let $\mathcal G_n$ denote the space of $n$-generated marked groups. We prove that, for every $n\ge 2$, there exist $2^{\aleph_0}$ non-atomic, $Out(F_n)$-invariant, mixing probability measures on $\mathcal G_n$. On the other hand, there are…

We study mixing properties of commutative groups of automorphisms acting on compact nilmanifolds. Assuming that every nontrivial element acts ergodically, we prove that such actions are mixing of all orders. We further show exponential…

We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups $G$…

We define the associated variety $ X_{M} $ of a module $ M $ over a finite-dimensional superalgebra $ {\mathfrak g} $, and show how to extract information about $ M $ from these geometric data. $ X_{M} $ is a subvariety of the cone $ X $ of…

In the classification theorems of Vinberg and Yakimova for commutative nilmanifolds, the relevant nilpotent groups have a very surprising analytic property. The manifolds are of the form $G/K = N \rtimes K/K$ where, in all but three cases,…

We say that two unitary or orthogonal representations of a finitely generated group $G$ are additive conjugates if they are intertwined by an additive map, which need not be continuous. We associate to each representation of $G$ a…

If M is a compact oriented manifold-with-boundary whose fundamental group is virtually nilpotent or Gromov-hyperbolic, we show that the higher signatures of M are oriented-homotopy invariants.

We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of ${\rm GL}_n(\mathbf{C})$ on the variety of $x$-nilpotent complex matrices and translate it to a representation-theoretic context. We obtain a criterion as…

Let M be a twisted interval bundle over a nonorientable hyperbolizable surface. Let X(M) be the PSL(2,C)-character variety of \pi_1(M). We examine the dynamics of the action of Out(\pi_1(M)) on X(M), and in particular, we find an open set…

We survey recent work on the dynamics of the outer automorphism group of a word hyperbolic group on spaces of (conjugacy classes of) representations ofthe group into a semi-simple Lie group G. All these results are motivated by the fact…

Let $G$ be a real semisimple Lie group with trivial centre and no compact factors. Given a conjugate pair of either real hyperbolic elements or unipotent elements $a$ and $b$ in $G$ we find a conjugating element $g \in G$ such that…

Given a $\mathbb Z^r$-action $\alpha$ on a nilmanifold $X$ by automorphisms and an ergodic $\alpha$-invariant probability measure $\mu$, we show that $\mu$ is the uniform measure on $X$, unless modulo finite index modification, one of the…

We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of ${\rm GL}_n(\mathbf{C})$, especially of the Borel subgroup $B$ and of the standard unipotent subgroup $U$ of the latter on the nilpotent cone of complex…

The space AH(M) of marked hyperbolic 3-manifold homotopy equivalent to a compact 3-manifold with boundary M sits inside the PSL_2(C)-character variety X(M) of \pi_1(M). We study the dynamics of the action of Out(\pi_1(M)) on both AH(M) and…