Related papers: Volume of representations and mapping degree

We survey some important results concerning the finite--dimensional representations of the loop algebra of a simple complex Lie algebra, and their twisted loop subalgebras. In particular, we review the parametrization and description of the…

Let $\mathcal{N}_g$ be the mapping class group of a non-orientable closed surface. We prove that the proper cohomological dimension, the proper geometric dimension, and the virtual cohomological dimension of $\mathcal{N}_g$ are equal…

Harmonic manifolds of hypergeometric type form a class of non-compact harmonic manifolds that includes rank one symmetric spaces of non-compact type and Damek-Ricci spaces. When normalizing the metric of a harmonic manifold of…

We outline the current state of knowledge regarding geometric inequalities of systolic type, and prove new results, including systolic freedom in dimension 4. Namely, every compact, orientable, smooth 4-manifold X admits metrics of…

Any sufficiently often differentiable curve in the orbit space $V/G$ of a real finite-dimensional orthogonal representation $G \to O(V)$ of a finite group $G$ admits a differentiable lift into the representation space $V$ with locally…

We extend the concept of renormalized volume for geometrically finite hyperbolic $3$-manifolds, and show that is continuous for geometrically convergent sequences of hyperbolic structures over an acylindrical 3-manifold $M$ with…

For closed and oriented hyperbolic surfaces, a formula of Witten establishes an equality between two volume forms on the space of representations of the surface in a semisimple Lie group. One of the forms is a Reidemeister torsion, the…

Let G be a subgroup of GL(V), where V is a finite dimensional vector space over a finite field of characteristic p >0. If det(g-1) = 0 for all g \in G then we call G a fixed-point subgroup of GL(V). Motivated in parallel by questions in…

Let $(M^n,g)$ be a complete Riemannian manifold which is not isometric to $\mathbb{R}^n$, has nonnegative Ricci curvature, Euclidean volume growth, and quadratic Riemann curvature decay. We prove that there exists a set $\mathcal{G}\subset…

The study of the volume of big line bundles on a complex projective manifold M has been one of the main veins in the recent interest in the asymptotic properties of linear series. In this article, we consider an equivariant version of this…

We study the spectrum of simplicial volume for closed manifolds with fixed fundamental group and relate the gap problem to rationality questions in bounded (co)homology. In particular, we show that in many cases this spectrum has a gap at…

We obtain a standard local presentation for a vector-valued multisymplectic form on a smooth manifold, generalizing the known proof for polysymplectic forms. We show that vector-valued multisymplectic forms on a finite-dimensional real…

Let M be a smooth compact connected oriented manifold of dimension at least two endowed with a volume form. We show that every homogeneous quasi-morphism on the identity component $Diff_0(M,vol)$ of the group of volume preserving…

In this paper we generalize a result in [1], showing that an arbitrary Riemannian symmetric space can be realized as a closed submanifold of a covering group of the Lie group defining the symmetric space. Some properties of the subgroups of…

This is an expository proof that, if $M$ is a compact $n$-manifold with no boundary, then the set of holonomies of strictly-convex real-projective structures on $M$ is a subset of $\operatorname{Hom}(\pi_1M,\operatorname{PGL}(n+1,\mathbb…

Does a space enjoying good finiteness properties admit an algebraic model with commensurable finiteness properties? In this note, we provide a rational homotopy obstruction for this to happen. As an application, we show that the maximal…

Given a Lie group $G$ we study the class $\M$ of proper metrizable $G$-spaces with metrizable orbit spaces, and show that any $G$-space $X \in \M$ admits a closed $G$-embedding into a convex $G$-subset $C$ of some locally convex linear…

As for the theory of maximal representations, we introduce the volume of a Zimmer's cocycle $\Gamma \times X \rightarrow \mbox{PO}^\circ(n, 1)$, where $\Gamma$ is a torsion-free (non-)uniform lattice in $\mbox{PO}^\circ(n, 1)$, with $n \geq…

Let $M^m$, with $m\geq 3$, be an $m$-dimensional complete noncompact manifold isometrically immersed in a Hadamard manifold $\bar M$. Assume that the mean curvature vector has finite $L^p$-norm, for some $2\leq p\leq m$. We prove that each…

Volume estimates of metric balls in manifolds find diverse applications in information and coding theory. In this paper, some new results for the volume of a metric ball in unitary group are derived via various tools from random matrix…