arXiv subjects

Alessio Savini

Publications and source records attributed to Alessio Savini.

32 records · Page 2Linked to original sources

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 $\mathbb{H}^3$.

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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.

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arXiv2021

Multiplicative constants and maximal measurable cocycles in bounded cohomology

Multiplicative constants are a fundamental tool in the study of maximal representations. In this paper we show how to extend such notion, and the associated framework, to measurable cocycles theory. As an application of this approach, we define and study the Cartan invariant for measurable $\textup{PU}(m,1)$-cocycles of complex hyperbolic lattices.

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arXiv2021

Superrigidity of maximal measurable cocycles of complex hyperbolic lattices

Let $Γ$ be a torsion-free lattice of $\text{PU}(p,1)$ with $p \geq 2$ and let $(X,μ_X)$ be an ergodic standard Borel probability $Γ$-space. We prove that any maximal Zariski dense measurable cocycle $σ: Γ\times X \longrightarrow \text{SU}(m,n)$ is cohomologous to a cocycle associated to a representation of $\text{PU}(p,1)$ into $\text{SU}(m,n)$, with $1 < m \leq n$. The proof follows the line of Zimmer' Superrigidity Theorem and requires the existence of a boundary map, that we prove in a much more general setting. As a consequence of our result, it cannot exist a maximal measurable cocycle with the above properties when $n\neq m$.

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arXiv2021

Equivariant maps for measurable cocycles with values into higher rank Lie groups

Let $G$ a semisimple Lie group of non-compact type and let $\mathcal{X}_G$ be the Riemannian symmetric space associated to it. Suppose $\mathcal{X}_G$ has dimension $n$ and it has no factor isometric to either $\mathbb{H}^2$ or $\text{SL}(3,\mathbb{R})/\text{SO}(3)$. Given a closed $n$-dimensional Riemannian manifold $N$, let $Γ=π_1(N)$ be its fundamental group and $Y$ its universal cover. Consider a representation $ρ:Γ\rightarrow G$ with a measurable $ρ$-equivariant map $ψ:Y \rightarrow \mathcal{X}_G$. Connell-Farb described a way to construct a map $F:Y\rightarrow \mathcal{X}_G$ which is smooth, $ρ$-equivariant and with uniformly bounded Jacobian. In this paper we extend the construction of Connell-Farb to the context of measurable cocycles. More precisely, if $(Ω,μ_Ω)$ is a standard Borel probability $Γ$-space, let $σ:Γ\times Ω\rightarrow G$ be a measurable cocycle. We construct a measurable map $F: Y \times Ω\rightarrow \mathcal{X}_G$ which is $σ$-equivariant, whose slices are smooth and they have uniformly bounded Jacobian. For such equivariant maps we define also the notion of volume and we prove a sort of mapping degree theorem in this particular context.

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arXiv2021

Parametrized Euler class and semicohomology theory

We extend Ghys' theory about semiconjugacy to the world of measurable cocycles. More precisely, given a measurable cocycle with values into $\text{Homeo}^+(\mathbb{S}^1)$, we can construct a $\text{L}^\infty$-parametrized Euler class in bounded cohomology. We show that such a class vanishes if and only if the cocycle can be lifted to $\text{Homeo}^+_{\mathbb{Z}}(\mathbb{R})$ and it admits an equivariant family of points. We define the notion of semicohomologous cocycles and we show that two measurable cocycles are semicohomologous if and only if they induce the same parametrized Euler class. Since for minimal cocycles, semicohomology boils down to cohomology, the parametrized Euler class is constant for minimal cohomologous cocycles. We conclude by studying the vanishing of the real parametrized Euler class and we obtain some results of elementarity.

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arXiv2021

Entropy rigidity for foliations by strictly convex projective manifolds

Let $N$ be a compact manifold with a foliation $\mathscr{F}_N$ whose leaves are compact strictly convex projective manifolds. Let $M$ be a compact manifold with a foliation $\mathscr{F}_M$ whose leaves are compact hyperbolic manifolds of dimension bigger than or equal to $3$. Suppose to have a foliation-preserving homeomorphism $f:(N,\mathscr{F}_N) \rightarrow (M,\mathscr{F}_M)$ which is $C^1$-regular when restricted to leaves. In the previous situation there exists a well-defined notion of foliated volume entropies $h(N,\mathscr{F}_N)$ and $h(M,\mathscr{F}_M)$ and it holds $h(M,\mathscr{F}_M) \leq h(N,\mathscr{F}_N)$. Additionally, if equality holds, then the leaves must be homothetic.

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arXiv2020

Algebraic hull of maximal measurable cocycles of surface groups into Hermitian Lie groups

Following the work of Burger, Iozzi and Wienhard for representations, in this paper we introduce the notion of maximal measurable cocycles of a surface group. More precisely, let $\mathbf{G}$ be a semisimple algebraic $\mathbb{R}$-group such that $G=\mathbf{G}(\mathbb{R})^\circ$ is of Hermitian type. If $Γ\leq L$ is a torsion-free lattice of a finite connected covering of $\text{PU}(1,1)$, given a standard Borel probability $Γ$-space $(Ω,μ_Ω)$, we introduce the notion of Toledo invariant for a measurable cocycle $σ:Γ\times Ω\rightarrow G$. The Toledo remains unchanged along $G$-cohomology classes and its absolute value is bounded by the rank of $G$. This allows to define maximal measurable cocycles. We show that the algebraic hull $\mathbf{H}$ of a maximal cocycle $σ$ is reductive and the centralizer of $H=\mathbf{H}(\mathbb{R})^\circ$ is compact. If additionally $σ$ admits a boundary map, then $H$ is of tube type and $σ$ is cohomologous to a cocycle stabilizing a unique maximal tube-type subdomain. This result is analogous to the one obtained for representations. In the particular case $G=\text{PU}(n,1)$ maximality is sufficient to prove that $σ$ is cohomologous to a cocycle preserving a complex geodesic. We conclude with some remarks about boundary maps of maximal Zariski dense cocycles.

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arXiv2020

A Matsumoto-Mostow result for Zimmer's cocycles of hyperbolic lattices

As for the theory of maximal representations, we introduce the volume of a Zimmer's cocycle $Γ\times X \rightarrow \mbox{PO}^\circ(n, 1)$, where $Γ$ is a torsion-free (non-)uniform lattice in $\mbox{PO}^\circ(n, 1)$, with $n \geq 3$, and $X$ is a suitable standard Borel probability $Γ$-space. Our numerical invariant extends the volume of representations for (non-)uniform lattices to measurable cocycles and in the uniform setting it agrees with the generalized version of the Euler number of self-couplings. We prove that our volume of cocycles satisfies a Milnor-Wood type inequality in terms of the volume of the manifold $Γ\backslash \mathbb{H}^n$. This invariant can be interpreted as a suitable multiplicative constant between bounded cohomology classes. This allows us to characterize maximal cocycles for being cohomologous to the cocycle induced by the standard lattice embedding via a measurable map $X \rightarrow \mbox{PO}(n, 1)$ with essentially constant sign. As a by-product of our rigidity result for the volume of cocycles, we give a new proof of the mapping degree theorem. This allows us to provide a complete characterization of maps homotopic to local isometries between closed hyperbolic manifolds in terms of maximal cocycles. In dimension $n = 2$, we introduce the notion of Euler number of measurable cocycles associated to closed surface groups. It extends the classic Euler number of representations and it agrees with the generalized version of the Euler number of self-couplings up to a multiplicative constant. We show a Milnor-Wood type inequality whose upper bound is given by the modulus of the Euler characteristic. This gives an alternative proof of the same result for the generalized version of the Euler number of self-couplings. Finally, we characterize maximal cocycles as those which are cohomologous to the one induced by a hyperbolization.

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arXiv2019

Asymptotically Moebius maps and rigidity for the hyperbolic plane

Let $S$ be a rank-one symmetric space of non-compact type and let $X$ be a $\text{CAT}(-1)$ space. A well-known result by Bourdon states that if a topological embedding $φ: \partial_\infty S \rightarrow \partial_\infty X$ respects cross ratios, that means $\text{cr}_S( ξ_0,η_0,ξ_1,η_1)=\text{cr}_X( φ(ξ_0),φ(η_0),φ(ξ_1),φ(η_1))$ for every $ξ_0,η_0,ξ_1,η_1 \in \partial_\infty S$, then $φ$ is induced by an isometric embedding of $S$ into $X$. We generalize this result when $S=\mathbb{H}^2$ is the real hyperbolic plane as it follows. Let $φ_k: \partial_\infty \mathbb{H}^2 \rightarrow \partial_\infty X$ be a sequence of continuous maps which are asymptotically Moebius, that means $\lim_{k \to \infty} \text{cr}_X(φ_k(ξ_0),φ_k(η_0),φ_k(ξ_1),φ_k(η_1))=\text{cr}_{\mathbb{H}^2}( ξ_0,η_0,ξ_1,η_1)$ for every $ξ_0,η_0,ξ_1,η_1 \in \partial_\infty \mathbb{H}^2$. Assume that the isometry group $\text{Isom}(X)$ acts transitively on triples of distinct points of $\partial_\infty X$. Then there must exists a sequence $(g_k)_{k \in \mathbb{N}}$, $g_k \in \text{Isom}(X)$ and a map $φ_\infty: \partial_\infty \mathbb{H}^2\rightarrow \partial_\infty X$ such that $\lim_{k \to \infty} g_kφ_k(ξ)=φ_\infty(ξ)$ for every $ξ\in \partial_\infty \mathbb{H}^2$ and $φ_\infty$ is induced by an isometric embedding of $\mathbb{H}^2$ into $X$.

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arXiv2018

Volume rigidity at ideal points of the character variety of hyperbolic 3-manifolds

Given the fundamental group $Γ$ of a finite-volume complete hyperbolic $3$-manifold $M$, it is possible to associate to any representation $ρ:Γ\rightarrow \text{Isom}(\mathbb{H}^3)$ a numerical invariant called volume. This invariant is bounded by the hyperbolic volume of $M$ and satisfies a rigidity condition: if the volume of $ρ$ is maximal, then $ρ$ must be conjugated to the holonomy of the hyperbolic structure of $M$. This paper generalizes this rigidity result by showing that if a sequence of representations of $Γ$ into $\text{Isom}(\mathbb{H}^3)$ satisfies $\lim_{n \to \infty} \text{Vol}(ρ_n) = \text{Vol}(M)$, then there must exist a sequence of elements $g_n \in \text{Isom}(\mathbb{H}^3)$ such that the representations $g_n \circ ρ_n \circ g_n^{-1}$ converge to the holonomy of $M$. In particular if the sequence $ρ_n$ converges to an ideal point of the character variety, then the sequence of volumes must stay away from the maximum. We conclude by generalizing the result to the case of $k$-manifolds and representations in $\text{Isom}(\mathbb H^m)$, where $m\geq k$.

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arXiv2018

Rigidity at infinity for lattices in rank-one Lie groups

Let $Γ$ be a non-uniform lattice in $PU(p,1)$ without torsion and with $p\geq2 $. We introduce the notion of volume for a representation $ρ:Γ\rightarrow PU(m,1)$ where $m \geq p$. We use this notion to generalize the Mostow--Prasad rigidity theorem. More precisely, we show that given a sequence of representations $ρ_n:Γ\rightarrow PU(m,1)$ such that $\lim_{n \to \infty} \text{Vol}(ρ_n) =\text{Vol}(M)$, then there must exist a sequence of elements $g_n \in PU(m,1)$ such that the representations $g_n \circ ρ_n \circ g_n^{-1}$ converge to a reducible representation $ρ_\infty$ which preserves a totally geodesic copy of $\mathbb{H}^p_\mathbb{C}$ and whose $\mathbb{H}^p_\mathbb{C}$-component is conjugated to the standard lattice embedding $i:Γ\rightarrow PU(p,1) < PU(m,1)$. Additionally, we show that the same definitions and results can be adapted when $Γ$ is a non-uniform lattice of $PSp(p,1)$ without torsion and for representations $ρ:Γ\rightarrow PSp(m,1)$, still mantaining the hypothesis $m \geq p \geq 2$.

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arXiv2017

The $ω$-Borel invariant for representations into $SL(n,\mathbb{C}_ω)$

Let $Γ$ be the fundamental group of a complete hyperbolic $3$-manifold $M$ with toric cusps. We define the $ω$-Borel invariant $β_n^ω(ρ_ω)$ associated to a representation $ρ_ω: Γ\rightarrow SL(n,\mathbb{C}_ω)$, where $\mathbb{C}_ω$ is a field which can be constructed as a quotient of a suitable subset of $\mathbb{C}^\mathbb{N}$ with the data of a non-principal ultrafilter $ω$ on $\mathbb{N}$ and a real divergent sequence $λ_l$ such that $λ_l \geq 1$. Since a sequence of $ω$-bounded representations $ρ_l$ into $SL(n,\mathbb{C})$ determines a representation $ρ_ω$ into $SL(n,\mathbb{C}_ω)$, for $n=2$ we study the relation between the invariant $β^ω_2(ρ_ω)$ and the sequence of Borel invariants $β_2(ρ_l)$. We conclude by showing that if a sequence of representations $ρ_l:Γ\rightarrow SL(2,\mathbb{C})$ induces a representation $ρ_ω:Γ\rightarrow SL(2,\mathbb{C}_ω)$ which determines a reducible action on the asymptotic cone $C_ω(\mathbb{H}^3,d/λ_l,O)$ with non-trivial length function, then it holds $β^ω_2(ρ_ω)=0$.

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arXiv2015

Fibered knots and links in lens spaces

We take advantage of the correspondence between fibered links, open book decompositions and contact structures on a closed connected 3-dimensional manifold to determine a mixed link diagram presentation for a particular fibered link $L$ in the lens space $L(p,q)$. Moreover, we construct a diagram for the lift of $L$ in the 3-sphere $S^3$ in order to show that, for $q=1$, $L$ is compatible with the standard contact structure of $L(p,1)$ when $p \leq -1$.

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