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Alessio Savini

Publications and source records attributed to Alessio Savini.

At least 19 recordsLinked to original sources
arXiv2026

Superrigidity for representations of transverse measured groupoids

For $i=1,\ldots,k$, let $\mathbf{G}_i$ be a connected, simply connected, semisimple algebraic group over some local field $\kappa_i$ of characteristic zero. Let $G_i=\mathbf{G}_i(\kappa_i)$ be the $\kappa_i$-points of $\mathbf{G}_i$ and denote by $G=\prod_{i=1}^k G_i$. If we assume that $G$ has higher rank and each factor has positive rank, given an ergodic transverse $G$-system $(X,\mu,Y)$, we prove a superrigidity phenomenon for Zariski dense representations of the transverse groupoid $(G \ltimes X)|_Y$ into either an almost simple or a reductive algebraic group.

math.DS↗
arXiv2025

Continuous cochains on Furstenberg boundaries and injectivity of the comparison map

Monod proved that any continuous cohomology of a semisimple Lie group $G$ can be represented by a measurable cocycle on the associated Furstenberg boundary, which we upgraded to an alternating cocycle. In the current paper we improve that result by showing that we can actually take a representing cocycle which is continuous on an explicit subset of generic tuples. We give an analogous result in the case of bounded cohomology. Finally, we exploit this characterization to prove the injectivity of the comparison map in degree $3$ for $\mathrm{Isom}^\circ(\mathbb{H}_{\mathbb{C}}^n)$, when $n \geq 2$, and in degree $4$ for $\mathrm{Isom}^\circ(\mathbb{H}^n_{\mathbb{R}})$, when $n \geq 2$.

math.GR↗
arXiv2025

Projections from Furstenberg boundaries onto maximal flats and barycenter maps

Let $G$ be a semisimple connected Lie group of non-compact type with finite center. Let $K<G$ be a maximal compact subgroup and $P<G$ be a minimal parabolic subgroup. For any pair $(F,x)$, where $F$ is a maximal flat in $G/K$ and $x \in G/P$ is opposite to the Weyl chambers determined by $F$, we define a projection $\Phi(F, x) \in F$ which is continuous and $G$-equivariant. Furthermore, if $q \geq 3$, we exhibit a $G$-equivariant continuous map defined on an open subset of full measure of the space of $q$-tuples of $(G/P)^q$ with image in $G/K$. When $G$ is the orientation preserving isometries of real hyperbolic space and $q = 3$, we recover the geometric barycenter of the corresponding ideal triangle. All our proofs are constructive.

math.GR↗
arXiv2025

Measurable bounded cohomology of $t$-discrete measured groupoids via resolutions

We define bounded cohomology of $t$-discrete measured groupoids with coefficients into measurable bundles of Banach spaces. Our approach via homological algebra extends the classic theory developed by Ivanov and by Monod. As a consequence, we show that the bounded cohomology of a $t$-discrete groupoid $\mathcal{G}$ can be computed using any amenable $\mathcal{G}$-space. In particular, we can compute bounded cohomology using strong boundaries.

math.AT↗
arXiv2024

Alternating cochains on Furstenberg boundaries and measurable cohomology

Nicolas Monod showed that the evaluation map $$H^*_m(G\curvearrowright G/P)\longrightarrow H^*_m(G)$$ between the measurable cohomology of the action of a connected semisimple Lie group $G$ on its Furstenberg boundary $G/P$ and the measurable cohomology of $G$ is surjective with a non-trivial kernel in all degrees below a constant depending on $G$ and less than or equal to the rank of $G$ plus $2$. When we were looking for explicit representatives of classes in this kernel, we were astonished to discover that some of these nontrivial classes have trivial alternation. In this paper, we refine Monod's result by identifying the non-alternating and alternating cohomology classes in this kernel. As a consequence, we show that $H^*_m(G)$ is isomorphic to the alternating measurable cohomology of $G$ acting on $G/P$ in all even degrees $$H^{2k}_{m,\mathrm{alt}}(G\curvearrowright G/P)\cong H^{2k}_m(G),$$ for a majority of Lie groups, namely those for which the longest element of the Weyl group acts as $-1$ on the Lie algebra of a maximal split torus $A$ in $G$.

math.GR↗
arXiv2024

Boundaries and equivariant maps for ergodic groupoids

We give a notion of boundary pair $(\mathcal{B}_-,\mathcal{B}_+)$ for measured groupoids which generalizes the one introduced by Bader and Furman \cite{BF14} for locally compact groups. In the case of a semidirect groupoid $\mathcal{G}=\Gamma \ltimes X$ obtained by a probability measure preserving action $\Gamma \curvearrowright X$ of a locally compact group, we show that a boundary pair is exactly $(B_- \times X, B_+ \times X)$, where $(B_-,B_+)$ is a boundary pair for $\Gamma$. For any measured groupoid $(\mathcal{G},\nu)$, we prove that the Poisson boundaries associated to the Markov operators generated by a probability measure equivalent to $\nu$ provide other examples of our definition. Following Bader and Furman \cite{BF:Unpub}, we define algebraic representability for an ergodic groupoid $(\mathcal{G},\nu)$. In this way, given any measurable representation $\rho:\mathcal{G} \rightarrow H$ into the $\kappa$-points of an algebraic $\kappa$-group $\mathbf{H}$, we obtain $\rho$-equivariant maps $\mathcal{B}_\pm \rightarrow H/L_\pm$, where $L_\pm=\mathbf{L}_\pm(\kappa)$ for some $\kappa$-subgroups $\mathbf{L}_\pm<\mathbf{H}$. In the particular case when $\kappa=\mathbb{R}$ and $\rho$ is Zariski dense, we show that $L_\pm$ must be minimal parabolic subgroups.

math.DS↗
arXiv2023

On the trivializability of rank-one cocycles with an invariant field of projective measures

Let $G$ be $\text{SO}^\circ(n,1)$ for $n \geq 3$ and consider a lattice $Γ< G$. Given a standard Borel probability $Γ$-space $(Ω,μ)$, consider a measurable cocycle $σ:Γ\times Ω\rightarrow \mathbf{H}(κ)$, where $\mathbf{H}$ is a connected algebraic $κ$-group over a local field $κ$. Under the assumption of compatibility between $G$ and the pair $(\mathbf{H},κ)$, we show that if $σ$ admits an equivariant field of probability measures on a suitable projective space, then $σ$ is trivializable. An analogous result holds in the complex hyperbolic case.

math.DS↗
arXiv2023

Kernels in measurable cohomology for transitive actions

Given a connected semisimple Lie group $G$, Monod has recently proved that the measurable cohomology of the $G$-action $H^*_m(G \curvearrowright G/P)$ on the Furstenberg boundary $G/P$, where $P$ is a minimal parabolic subgroup, maps surjectively on the measurable cohomology of $G$ through the evaluation on a fixed basepoint. Additionally, the kernel of this map depends entirely on the invariant cohomology of a maximal split torus. In this paper we show a similar result for a fixed subgroup $L<P$ such that the stabilizer of almost every pair of points in $G/L$ is compact. More precisely, we show that the cohomology of the $G$-action $H^p_m(G \curvearrowright G/L)$ maps surjectively onto $H^p_m(G)$ with a kernel isomorphic to $H^{p-1}_m(L)$. Examples of such groups are given either by any term of the derived series of the unipotent radical $N$ of $P$ or by a maximal split torus $A$. We conclude the paper by computing explicitly some cocycles on quotients of $\mathrm{SL}(2,\mathbb{K})$ for $\mathbb{K}=\mathbb{R}, \mathbb{C}$.

math.GR↗
arXiv2023

Measurable bounded cohomology of measured groupoids

We introduce the notion of measurable bounded cohomology for measured groupoids, extending continuous bounded cohomology of locally compact groups. We show that the measurable bounded cohomology of the semidirect groupoid associated to a measure class preserving action of a locally compact group $G$ on a standard Borel space is isomorphic to the continuous bounded cohomology of $G$ with twisted coefficients. We also prove the invariance of measurable bounded cohomology under similarity. As an application, we compare the bounded cohomology of (weakly) orbit equivalent actions and of measure equivalent groups. In this way we recover an isomorphism in bounded cohomology similar to one proved by Monod and Shalom. Other relevant consequences are related to the cohomological vanishing for actions of the Thompson group $F$, of higher rank lattices and of lattices in products of locally compact groups. We obtain a variant of the Eckmann-Shapiro isomorphism for transitive actions. In the case of a higher rank simple Lie group, we show that the cohomology of the action is actually determined by the usual cohomology of a suitable lattice. For amenable groupoids, we prove that the measurable bounded cohomology is trivial. This generalizes previous results by Monod, Anantharaman-Delaroche and Renault, and Blank.

math.DS↗
arXiv2022

Bounded Cohomology Classes of Exact Forms

On negatively curved compact manifolds, it is possible to associate to every closed form a bounded cocycle - hence a bounded cohomology class - via integration over straight simplices. The kernel of this map is contained in the space of exact forms. We show that in degree 2 this kernel is trivial, in contrast with higher degree. In other words, exact non-zero $2$-forms define non-trivial bounded cohomology classes. This result is the higher dimensional version of a classical theorem by Barge and Ghys for surfaces. As a consequence, one gets that the second bounded cohomology of negatively curved manifolds contains an infinite dimensional space, whose classes are explicitly described by integration of forms. This also showcases that some recent results by Marasco (arXiv:2202.04419, arXiv:2209.00560) can be applied in higher dimension to obtain new non-trivial results on the vanishing of certain cup products and Massey products. Some other applications are discussed.

math.GT↗
arXiv2022

Integrable tautness of isometries of complex hyperbolic spaces

Consider $n \geq 2$. In this paper we prove that the group $\text{PU}(n,1)$ is $1$-taut. This result concludes the study of $1$-tautness of rank-one Lie groups of non-compact type. Additionally the tautness property implies a classification of finitely generated groups which are $\text{L}^1$-measure equivalent to lattices of $\text{PU}(n,1)$. More precisely, we show that $\text{L}^1$-measure equivalent groups must be extensions of lattices of $\text{PU}(n,1)$ by a finite group.

math.GT↗
arXiv2022

Some explicit cocycles on the Furstenberg boundary for products of isometries of hyperbolic spaces and $\mathrm{SL}(3,\mathbb{K})$

Nicolas Monod showed that the evaluation map $H^*_m(G\curvearrowright G/P)\longrightarrow H^*_m(G)$ between the measurable cohomology of the action of a connected semisimple Lie group $G$ on its Furstenberg boundary $G/P$ and the measurable cohomology of $G$ is surjective with a kernel that can be entirely described in terms of invariants in the cohomology of a maximal split torus $A<G$. In a recent paper we refine Monod's result and show in particular that the cohomology of non-alternating cocycles on $G/P$ is in general not trivial and lies in the kernel of the evaluation. In this paper we describe explicitly such non-alternating and alternating cocycles on $G/P$ in low degrees when $G$ is either a product of isometries of real hyperbolic spaces or $G=\mathrm{SL}(3,\mathbb{K})$, where $\mathbb{K}$ is either the real or the complex field. As a consequence, we deduce that the comparison map $H^*_{m,b}(G)\rightarrow H^*_m(G)$ from the measurable bounded cohomology is injective in degree $3$ for nontrivial products of isometries of hyperbolic spaces. We get also another proof of the injectivity for $G=\mathrm{SL}(3,\mathbb{K})$, when $\mathbb{K}$ is either the real field or the complex one.

math.GR↗
arXiv2022

Orbital cohomology and Kahler rigidity

In the late $70$'s Feldman and Moore defined the cohomology associated to a countable equivalence relation with coefficients in an Abelian Polish group. When the equivalence relation is the orbital one, that is it is induced by a measure preserving action of a countable group $Γ$ on a standard Borel probability space $(X,μ)$, it still makes sense to consider the Feldmann-Moore $1$-cohomology with $G$-coefficients, where this time $G$ can be any topological group. The latter cohomology, denoted by $H^1(Γ\curvearrowright X;G)$, is very misterious and hard to compute, except for some exceptional cases. In this expository paper we are going to focus our attention on the particular case when $Γ$ is a finitely generated group and $G$ is a Hermitian Lie group. We are going to give some recent rigidity results in this context and we will see how those results can be used to say something relevant about (some subsets of) the orbital cohomology.

math.DS↗
arXiv2022

A Note on elementarity of virtual dendro-morphisms for higher rank lattices

Let $Γ$ be a discrete countable group and let $(Ω,μ)$ be an ergodic standard Borel probability $Γ$-space. Given any non-elementary virtual dendro-morphism (that is a measurable cocycle in the automorphism group of a dendrite), we construct a unitary representation $V$ with no invariant vectors such that $\text{H}^2_b(Γ;V)$ contains a non-zero class. As a consequence, all virtual dendro-morphisms of a higher rank lattice must be elementary.

math.DS↗
arXiv2021

Borel invariant for measurable cocycles of 3-manifold groups

We introduce the notion of pullback along a measurable cocycle and we use it to extend the Borel invariant studied by Bucher, Burger and Iozzi to the world of measurable cocycles. The Borel invariant is constant along cohomology classes and has bounded absolute value. This allows to define maximal cocycles. We conclude by proving that maximal cocycles are actually trivializable to the restriction of the irreducible representation.

math.GT↗
arXiv2021

Rigidity at infinity for the Borel function of the tetrahedral reflection lattice

If $Γ$ is the fundamental group of a complete finite volume hyperbolic $3$-manifold, Guilloux conjectured that the Borel function on the $\text{PSL}(n,\mathbb{C})$-character variety of $Γ$ should be rigid at infinity, that is it should stay bounded away from its maximum at ideal points. In this paper we prove Guilloux's conjecture in the particular case of the reflection group associated to a regular ideal tetrahedron of $\mathbb{H}^3$.

math.GT↗
arXiv2021

Natural maps for measurable cocycles of compact hyperbolic manifolds

Let $\text{G}(n)$ be equal either to $\text{PO}(n,1),\text{PU}(n,1)$ or $\text{PSp}(n,1)$ and let $Γ\leq \text{G}(n)$ be a uniform lattice. Denote by $\mathbb{H}^n_K$ the hyperbolic space associated to $\text{G}(n)$, where $K$ is a division algebra over the reals of dimension $d=\dim_{\mathbb{R}} K$. Assume $d(n-1) \geq 2$. In this paper we generalize natural maps to measurable cocycles. Given a standard Borel probability $Γ$-space $(X,μ_X)$, we assume that a measurable cocycle $σ:Γ\times X \rightarrow \text{G}(m)$ admits an essentially unique boundary map $ϕ:\partial_\infty \mathbb{H}^n_K \times X \rightarrow \partial_\infty \mathbb{H}^m_K$ whose slices $ϕ_x:\mathbb{H}^n_K \rightarrow \mathbb{H}^m_K$ are atomless for almost every $x \in X$. Then, there exists a $σ$-equivariant measurable map $F: \mathbb{H}^n_K \times X \rightarrow \mathbb{H}^m_K$ whose slices $F_x:\mathbb{H}^n_K \rightarrow \mathbb{H}^m_K$ are differentiable for almost every $x \in X$ and such that $\text{Jac}_a F_x \leq 1$ for every $a \in \mathbb{H}^n_K$ and almost every $x \in X$. The previous properties allow us to define the natural volume $\text{NV}(σ)$ of the cocycle $σ$. This number satisfies the inequality $\text{NV}(σ) \leq \text{Vol}(Γ\backslash \mathbb{H}^n_K)$. Additionally, the equality holds if and only if $σ$ is cohomologous to the cocycle induced by the standard lattice embedding $i:Γ\rightarrow \text{G}(n) \leq \text{G}(m)$, modulo possibly a compact subgroup of $\text{G}(m)$ when $m>n$. Given a continuous map $f:M \rightarrow N$ between compact hyperbolic manifolds, we also obtain an adaptation of the mapping degree theorem to this context.

math.GT↗