The problems of conjugacy rigidity and of entropy rigidity have a long history, with major breakthroughs in the last twenty years, for negatively curved compact manifolds and symmetric spaces. On the other hand, little is known for finite volume manifolds. I will survey the state of the art of these problems, and explain what can be generalized to finite volume manifolds, pointing out the main difficulties that arise in the non-compact case.
SEMINARI DI GEOMETRIA
Kazhdan's property (T) is a strong negation of amenability, relevant in the study of algebraic properties of groups, in the construction of expander graphs, in dynamics and in connection to the Baum-Connes conjectures. Various strengthened versions of property (T) have been formulated in recent years.
The Drinfeld double of a finite dimensional Hopf algebra is a quasi-triangular Hopf algebra with the canonical element as the universal R-matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum invariant of framed links is constructed using a ribbon Hopf algebra. In that construction, a copy of the universal R-matrix is attached to each crossing, and invariance under the Reidemeister III move is shown by the quantum Yang-Baxter equation of the universal R-matrix. On the other hand, R.
A classical theorem of Gromov states that the Betti numbers, i.e. the size of the free part of the homology groups, of negatively curved manifolds are bounded by the volume. We extend this theorem to the torsion part of the homology in all dimensions d > 3. From Gromov’s work it is known that in dimension 3 the size of torsion homology cannot be bounded in terms of the volume. In dimension 4 we give a somewhat precise estimate for the number of negatively curved manifolds of finite volume, up to homotopy, and in dimension d > 4 up to homeomorphism.
Consider a group G action on a metric space X by isometries, and subgroups M_1,..., M_l. One interesting and important problem is to study conditions under which non-emptyness of each M_i-fixed points ensures that of global (G-)fixed points. One extreme case is where G is a simple Lie group, X is an origin-excluded Hilbert space, and the action is given by a (strongly continuous) unitary representation. Then, the Howe--Moore property, based on the Mautner phenomenon, implies that for each(!) non-compact closed subgroup M, the existence of M-fixed points suffices that of G-fixed points.
Le rappresentazioni massimali sono le famiglie di sottogruppi di
In this work we characterize the subsets of R^n that are images of Nash maps f : R^m → R^n. We prove Shiota’s conjecture and show that a subset S ⊂ R^n is the image of a Nash map f : R^m → R^n if and only if S is semialgebraic, pure dimensional of dimension d ≤ m and there exists an analytic path α : [0, 1] → S whose image meets all the connected components of the set of regular points of S. Given a semialgebraic set S ⊂ R^n satisfying the previous properties, we provide a theoretical strategy to construct (after Nash approximation) a Nash map whose image is the semialgebraic set S.
Nel suo lavoro fondazionale Mess ha mostrato come associare a superfici di tipo spazio nello spazio Anti de Sitter deformazioni che preservano l’ area di metriche iperboliche su superfici. Ad esempio superfici plissettate nello spazio Anti de Sitter corrispondono attraverso questa costruzione ai terremoti nel senso di Thurston. Piu’ recentemente in un lavoro con Schlenker abbiamo studiato le deformazioni associate a superfici massime, dette mappe minimali Lagrangiane, e con Schlenker e Mondello le deformazioni
associate a superfici a curvatura costante, dette k-landslides.
Un risultato attribuito a Fricke-Klein afferma che i punti di una
componente connessa dello spazio Rep(S,PSL(2,R)) delle classi di