Alessandra Iozzi

LIMIT SETS OF DISCRETE GROUPS OF ISOMETRIES OF EXOTIC HYPERBOLIC SPACES

Let Γ be a geometrically finite discrete group of isometries of hyperbolic space HF , where F = R, C, H or O (in which case n = 2). We prove that the critical exponent of Γ equals the Hausdorff dimension of the limit sets Λ(Γ) and that the smallest eigenvalue of the Laplacian acting on square integrable functions is a quadratic function of either of them (when they are sufficiently large). A generalization of Hopf ergodicity theorem for the geodesic flow with respect to the Bowen-Margulis measure is also proven.

The median class and superrigidity of actions on CAT(0) cube complexes

We define a bounded cohomology class, called the {\em median class}, in the second bounded cohomology -- with appropriate coefficients --of the automorphism group of a finite dimensional CAT(0) cube complex X. The median class of X behaves naturally with respect to taking products and appropriate subcomplexes and defines in turn the {\em median class of an action} by automorphisms of X. We show that the median class of a non-elementary action by automorphisms does not vanish and we show to which extent it does vanish if the action is elementary. We obtain as a corollary a superrigidity result and show for example that any irreducible lattice in the product of at least two locally compact connected groups acts on a finite dimensional CAT(0) cube complex X with a finite orbit in the Roller compactification of X. In the case of a product of Lie groups, the Appendix by Caprace allows us to deduce that the fixed point is in fact inside the complex X. In the course of the proof we construct a \Gamma-equivariant measurable map from a Poisson boundary of \Gamma with values in the non-terminating ultrafilters on the Roller boundary of X.

A Dual Interpretation of the Gromov–Thurston Proof of Mostow Rigidity and Volume Rigidity for Representations of Hyperbolic Lattices

We use bounded cohomology to define a notion of volume of an \(\operatorname{SO}(n,1)\)-valued representation of a lattice \(\varGamma<\operatorname{SO}(n,1)\) and, using this tool, we give a complete proof of the volume rigidity theorem of Francaviglia and Klaff (Geom. Dedicata 117, 111–124 (2006)) in this setting. Our approach gives in particular a proof of Thurston’s version of Gromov’s proof of Mostow Rigidity (also in the non-cocompact case), which is dual to the Gromov–Thurston proof using the simplicial volume invariant.

Isometric embeddings in bounded cohomology

This paper is devoted to the construction of norm-preserving maps between bounded cohomology groups. For a graph of groups with amenable edge groups we construct an isometric embedding of the direct sum of the bounded cohomology of the vertex groups in the bounded cohomology of the fundamental group of the graph of groups. With a similar technique we prove that if (X,Y) is a pair of CW-complexes and the fundamental group of each connected component of Y is amenable, the isomorphism between the relative bounded cohomology of (X,Y) and the bounded cohomology of X in degree at least 2 is isometric. As an application we provide easy and self-contained proofs of Gromov Equivalence Theorem and of the additivity of the simplicial volume with respect to gluings along \pi_1-injective boundary components with amenable fundamental group.

Equivariant embeddings of trees into hyperbolic spaces

We study isometric actions of tree automorphism groups on the infinite-dimensional hyperbolic spaces. On the one hand, we exhibit a general one-parameter family of such representations and analyse the corresponding equivariant embeddings of the trees, showing that they are convex-cocompact and asymptotically isometric. On the other hand, focusing on the case of sufficiently transitive groups of automorphisms of locally finite trees, we classify completely all irreducible representations by isometries of hyperbolic spaces. It turns out that in this case our one-parameter family exhausts all non-elementary representations.

Bounded cohomology and totally real subspaces in complex hyperbolic geometry

We characterize representations of finitely generated discrete groups into (the connected component of) the isometry group of a complex hyperbolic space via the pullback of the bounded Kähler class.

Tessellations of homogeneous spaces of classical groups of real rank two

Let H be a closed, connected subgroup of a connected, simple Lie group G with finite center. The homogeneous space G/H has a tessellation if there is a discrete subgroup Γ of G, such that Γ acts properly discontinuously on G/H, and the double-coset space Γ\G/H is compact. Note that if either H or G/H is compact, then G/H has a tessellation; these are the obvious examples.It is not difficult to see that if G has real rank one, then only the obvious homogeneous spaces have tessellations. Thus, the first interesting case is when G has real rank two. In particular, Kulkarni and Kobayashi constructed examples that are not obvious when G=SO(2, 2n)° or SU(2, 2n). Oh and Witte constructed additional examples in both of these cases, and obtained a complete classification when G=SO(2, 2n)°. We simplify the work of Oh-Witte, and extend it to obtain a complete classification when G=SU(2, 2n). This includes the construction of another family of examples.The main results are obtained from methods of Benoist and Kobayashi: we fix a Cartan decomposition G=KA+K, and study the intersection (KHK)∩A+. Our exposition generally assumes only the standard theory of connected Lie groups, although basic properties of real algebraic groups are sometimes also employed; the specialized techniques that we use are developed from a fairly elementary level.

On order-preserving representations

In this article we introduce order preserving representations of fundamental groups of surfaces into Lie groups with bi-invariant orders. By relating order preserving representations to weakly maximal representations, introduced in arXiv:1305.2620, we show that order preserving representations into Lie groups of Hermitian type are faithful with discrete image and that the set of order preserving representations is closed in the representation variety. For Lie groups of Hermitian type whose associated symmetric space is of tube type we give a geometric characterization of these representations in terms of the causal structure on the Shilov boundary.

Algebraic hulls and smooth orbit equivalence

Let (Mi,,ui), i = 1, 2, be two manifolds with quasi-invariant measures, and let Hi c Diff(Mi) be connected Lie groups. If there is a measure class preserving diffeomorphism 0: M1 - > M2 which is a bijection of Hl-orbits and H2-orbits then we say that the actions are smoothly orbit equivalent. If the Hi-actions determine foliations g on the manifolds MiS then the map 0 is just a diffeomorphism between the foliations. These phenomena, and the corresponding phenomena arising in situations in which the (Mi, Ai) are just Borel (or topological) Hi-spaces and the map 0 is a measure class preserving Borel isomorphism (or homeomorphism), have been studied, using a variety of techniques, independently by several authors [B, CFW, D1, D2, K, PnZ, W1, W2, Z2, Z4]. Most of the results obtained require some strengthening of the hypotheses, such as finiteness and invariance of the measures and amenability or semisimplicity (in higher rank) of the groups acting. One of the results that we want to describe in this paper fits in this geometric setting and provides an obstruction, in terms of a geometric invariant of the actions, to the foliations being diffeomorphic. Recall that, if H acts ergodically and by diffeomorphisms on the n-dimensional manifold M, the algebraic hull of the H-action is the unique (up to conjugacy) smallest algebraic subgroup L c GL(n, R) such that there exists a measurable H-invariant reduction to L of the frame bundle on M on which H acts by automorphisms. (For an analytic definition see §2 and for the general context see [Z5, 9.2].) Then the normal algebraic hull of the H-action on M will be the projection of L in the direction normal to the orbits.

The Moduli Space of Boundary Compactifications of SL (2, R)

In an earlier paper, the authors introduced the notion of a boundary compactification of SL(2, R) and SL(2, C), a normal projective embedding of PSL2 arising as the Zariski closure of an orbit in (P1)n under the diagonal action of SL2. Here the moduli space of such boundary compactifications of SL(2, R) is shown to be a contractible hyperbolic orbifold, by using the Schwarz–Christoffel transformation to identify it with a quotient of the moduli space of equi-angular planar polygons.