arXiv subjects
Publications and source records attributed to Gabriele Viaggi.
We develop criteria to distinguish the homotopy types of Gromov-Thurston manifolds. Our approach is based on a description of their fundamental groups as virtual Dehn fillings of relatively hyperbolic groups.
We prove uniform linear bounds on the volume variation under drilling and filling operations on finite volume hyperbolic 3-manifolds.
We introduce robust families of submanifolds for a linear Lie group $G$. We show that they give rise to geometric subspaces of the representation space ${\rm Hom}(\Gamma,G)$. As an application, we give a unified short proof of results of Beyrer and Kassel and of Benoist and Koszul about the existence of higher Teichm\"uller components for $G={\rm SO}(p,q+1),{\rm SL}(p+1,\mathbb{R})$. Being based on very general principles, our approach might be suited for finding geometric components in various ${\rm Hom}(\Gamma,G)$.
We give a formula with explicit constants relating the subsurface projection $d_Y(\nu^-,\nu^+)$ of the end invariants $\nu^-,\nu^+$ of a hyperbolic 3-manifold $Q$ diffeomorphic to $S\times\mathbb{R}$ and the length of the geodesic representative in $Q$ of the multicurve $\partial Y$. This makes effective and computable the large projections versus short curves relation proved by Minsky. We give an application to closed hyperbolic 3-manifolds fibering over the circle providing a geometric analog of the uniform projection bound in fibered faces of Minsky and Taylor.
We consider 3-manifolds given as Heegaard splittings $M=H^-\cup_\Sigma H^+$ with the aim to describe the hyperbolic metric of $M$ under topological conditions on the splitting guaranteeing that the manifold is hyperbolic. In particular, given a suitable "sufficiently incompressible" curve $\gamma\subset\Sigma$, we show (without appealing to Geometrization) that $M$ is hyperbolic and we compute the length of $\gamma$ in terms of the projection coefficients of the disk sets, up to a uniform multiplicative error.
We study the geometry of hyperconvex representations of surface groups in ${\rm PSL}(d,\mathbb{C})$ and their deformation spaces: We produce a natural holomorphic extension of the classical Ahlfors--Bers map to a product of Teichm\"uller spaces of a canonical Riemann surface lamination and prove that the limit set of a hyperconvex representation in the full flag space has Hausdorff dimension 1 if and only if the representation is conjugate in ${\rm PSL}(d,\mathbb{R})$.
We study the geometry of hyperconvex representations of hyperbolic groups in ${\rm PSL}(d,\mathbb{C})$ and establish two structural results: a group admitting a hyperconvex representation is virtually isomorphic to a Kleinian group, and its hyperconvex limit set in the appropriate flag manifold has Hausdorff dimension strictly smaller than $2$.
We investigate properties of the pseudo-Riemannian volume, entropy, and diameter for convex cocompact representations $\rho : \Gamma \to \mathrm{SO}(p,q+1)$ of closed $p$-manifold groups. In particular: We provide a uniform lower bound of the product entropy times volume that depends only on the geometry of the abstract group $\Gamma$. We prove that the entropy is bounded from above by $p-1$ with equality if and only if $\rho$ is conjugate to a representation inside ${\rm S}({\rm O}(p,1)\times{\rm O}(q))$, which answers affirmatively to a question of Glorieux and Monclair. Lastly, we prove finiteness and compactness results for groups admitting convex cocompact representations with bounded diameter.
We consider the family of Heegaard splittings of genus $g$ at least two which are defined via a gluing map that is the $n$-th power of the Dehn twist along a curve that satisfies a natural topological assumption, namely pared acylindricity. We show that if $n$ is at least 14, then the Heegaard splitting has a hyperbolic metric for which the simple closed curve defining the Dehn twist is a closed geodesic of length at least $1.24/(n^2g)$ and at most $37.5/n^2$
In this article we construct many examples of properly convex irreducible domains divided by Zariski dense relatively hyperbolic groups in every dimension at least 3. This answers a question of Benoist. Relative hyperbolicity and non-strict convexity are captured by a family of properly embedded cones (convex hulls of points and ellipsoids) in the domain. Our construction is most flexible in dimension 3 where we give a purely topological criterion for the existence of a large deformation space of geometrically controlled convex projective structures with totally geodesic boundary on a compact 3-manifold.
We establish a link between the behavior of length functions on Teichm\"uller space and the geometry of certain anti de Sitter 3-manifolds. As an application, we give new purely anti de Sitter proofs of results of Teichm\"uller theory such as (strict) convexity of length functions along shear paths and geometric bounds on their second variation along earthquakes. Along the way, we provide shear-bend coordinates for Mess' anti de Sitter 3-manifolds.
We study maximal representations of surface groups $\rho:\pi_1(\Sigma)\to\mathrm{SO}_0(2,n+1)$ via the introduction of $\rho$-invariant pleated surfaces inside the pseudo-Riemannian space $\mathbb{H}^{2,n}$ associated to maximal geodesic laminations of $\Sigma$. We prove that $\rho$-invariant pleated surfaces are always embedded, acausal, and possess an intrinsic pseudo-metric and a hyperbolic structure. We describe the latter by constructing a shear cocycle from the cross ratio naturally associated to $\rho$. The process developed to this purpose applies to a wide class of cross ratios, including examples arising from Hitchin and $\Theta$-positive representations in $\mathrm{SO}(p,q)$. We also show that the length spectrum of $\rho$ dominates the ones of $\rho$-invariant pleated surfaces, with strict inequality exactly on curves that intersect the bending locus. We observe that the canonical decomposition of a $\rho$-invariant pleated surface into leaves and plaques corresponds to a decomposition of the Guichard-Wienhard domain of discontinuity of $\rho$ into standard fibered blocks, namely triangles and lines of photons. Conversely, we give a concrete construction of photon manifolds fibering over hyperbolic surfaces by gluing together triangles of photons. The tools we develop allow to recover various results by Collier, Tholozan, and Toulisse on the (pseudo-Riemannian) geometry of $\rho$ and on the correspondence between maximal representations and fibered photon manifolds through a constructive and geometric approach, bypassing the use of Higgs bundles.
We show that for every $g\geq 2$ there exists a number $c(g)>0$ such that the smallest positive eigenvalue of a random closed 3-manifold $M$ of Heegaard genus $g$ is at most $c(g)/{\rm vol}(M)^2$.
We provide two constructions of hyperbolic metrics on 3-manifolds with Heegaard splittings that satisfy certain topological conditions, which both apply to random Heegaard splittings with asymptotic probability 1. These constructions provide a lot of control on the resulting metric, allowing us to prove various results about the coarse growth rate of geometric invariants, such as diameter and injectivity radius, and about arithmeticity and commensurability in families of random 3-manifolds. For example, we show that the diameter of a random Heegaard splitting grows coarsely linearly in the length of the associated random walk. The constructions only use tools from the deformation theory of Kleinian groups, that is, we do not rely on the solution of the Geometrization Conjecture by Perelman. In particular, we give a proof of Maher's result that random 3-manifolds are hyperbolic that bypasses Geometrization.
We prove a law of large numbers for the volumes of families of random hyperbolic mapping tori and Heegaard splittings providing a sharp answer to a conjecture of Dunfield and Thurston.