Anosov representations, strongly convex cocompact groups and weak eigenvalue gaps

Anosov representations of hyperbolic groups form a rich class of representations that are closely related to geometric structures on closed manifolds. Any Anosov representation $\rho :\Gamma \to G$ admits cocompact domains of discontinuity in flag varieties $G/Q$ [GW12, KLP18] endowing the compact quotient manifolds $M_\rho $ with a $(G,G/Q)$ –structure. In general, the topology of $M_\rho $ can be quite complicated. In this article, we will focus on the special case when $\Gamma $ is a the fundamental group of a closed (real or complex) hyperbolic manifold N and $\rho $ is a deformation of a (twisted) lattice embedding $\Gamma \to \mathrm {Isom}^\circ (\mathbb {H}_{\mathbb {K}}) \to G$ through Anosov representations. In this case, we prove that $M_\rho $ is a smooth fiber bundle over N, and we describe the structure group of this bundle and compute its invariants. This theorem applies in particular to most representations in higher rank Teichmüller spaces, as well as convex divisible representations, AdS-quasi-Fuchsian representations and $\mathbb {H}_{p,q}$ –convex cocompact representations. Even when $M_\rho \to N$ is a fiber bundle, it is often very difficult to determine the fiber. In the second part of the paper, we focus on the special case when N is a surface, $\rho $ a quasi-Hitchin representation into $\mathrm {Sp}(4,{\mathbb C})$ , and $M_\rho $ carries a $(\mathrm {Sp}(4,{{\mathbb C}}),\mathrm {Lag}({{\mathbb C}}^4))$ –structure. We show that in this case the fiber is homeomorphic to ${{\mathbb C}}\mathbb {P}^2 \# \overline {{{\mathbb C}}\mathbb {P}}^2$ . This fiber bundle $M_\rho \to N$ is of particular interest in the context of possible generalizations of Bers’ double uniformization theorem in the context of higher rank Teichmüller spaces, since for Hitchin-representations it contains two copies of the locally symmetric space associated to $\rho (\Gamma )$ . Our result uses the classification of smooth $4$ –manifolds, the study of the $\mathrm {SL}(2, {{\mathbb C}})$ –orbits of $\mathrm {Lag}({{\mathbb C}}^4)$ and the identification of $\mathrm {Lag}({{\mathbb C}}^4)$ with the space of (unlabelled) regular ideal hyperbolic tetrahedra and their degenerations.

Anosov representations of word hyperbolic groups into higher-rank semisimple Lie groups are representations with finite kernel and discrete image that have strong analogies with convex cocompact representations into rank-one Lie groups. However, the most naive analogy fails: generically, Anosov representations do not act properly and cocompactly on a convex set in the associated Riemannian symmetric space. We study representations into projective indefinite orthogonal groups $$\mathrm {PO}(p,q)$$ by considering their action on the associated pseudo-Riemannian hyperbolic space $$\mathbb {H}^{p,q-1}$$ in place of the Riemannian symmetric space. Following work of Barbot and Merigot in anti-de Sitter geometry, we find an intimate connection between Anosov representations and a natural notion of convex cocompactness in this setting.

In this paper we show that many projective Anosov representations act convex cocompactly on some properly convex domain in real projective space. In particular, if a non-elementary word hyperbolic group is not commensurable to a non-trivial free product or the fundamental group of a closed hyperbolic surface, then any projective Anosov representation of that group acts convex cocompactly on some properly convex domain in real projective space. We also show that if a projective Anosov representation preserves a properly convex domain, then it acts convex cocompactly on some (possibly different) properly convex domain. We then give three applications. First, we show that Anosov representations into general semisimple Lie groups can be defined in terms of the existence of a convex cocompact action on a properly convex domain in some real projective space (which depends on the semisimple Lie group and parabolic subgroup). Next, we prove a rigidity result involving the Hilbert entropy of a projective Anosov representation. Finally, we prove a rigidity result which shows that the image of the boundary map associated to a projective Anosov representation is rarely a $C^2$ submanifold of projective space. This final rigidity result also applies to Hitchin representations.

We describe multiple correlations of Jordan and Cartan spectra for any finite number of Anosov representations of a finitely generated group. This extends our previous work on correlations of length and displacement spectra for rank one convex cocompact representations. Examples include correlations of the Hilbert length spectra for convex projective structures on a closed surface as well as correlations of eigenvalue gaps and singular value gaps for Hitchin representations. We relate the correlation problem to the counting problem for Jordan and Cartan projections of an Anosov subgroup with respect to a family of carefully chosen truncated hypertubes, rather than in tubes as in our previous work. Hypertubes go to infinity in a linear subspace of directions, while tubes go to infinity in a single direction and this feature presents a novel difficulty in this higher rank correlation problem.

Using the thermodynamic formalism, we introduce a notion of intersection for projective Anosov representations, show analyticity results for the intersection and the entropy, and rigidity results for the intersection. We use the renormalized intersection to produce an Out $${(\Gamma)}$$ -invariant Riemannian metric on the smooth points of the deformation space of irreducible, generic, projective Anosov representations of a word hyperbolic group $${\Gamma}$$ into $${\mathsf{SL}_m(\mathbb{R})}$$ . In particular, we produce mapping class group invariant Riemannian metrics on Hitchin components which restrict to the Weil–Petersson metric on the Fuchsian loci. Moreover, we produce $${{\rm Out}(\Gamma)}$$ -invariant metrics on deformation spaces of convex cocompact representations into $${\mathsf{PSL}_2(\mathbb{C})}$$ and show that the Hausdorff dimension of the limit set varies analytically over analytic families of convex cocompact representations into any rank 1 semi-simple Lie group.

Anosov representations give a higher-rank analogue of convex cocompactness in a rank-one Lie group which shares many of its good geometric and dynamical properties; geometric finiteness in rank one may be seen as a controlled weakening of convex cocompactness to allow for isolated failures of hyperbolicity. We introduce relatively dominated representations as a relativization of Anosov representations, or in other words a higher-rank analogue of geometric finiteness. We prove that groups admitting relatively dominated representations must be relatively hyperbolic, that these representations induce limit maps with good properties, provide examples, and draw connections to work of Kapovich–Leeb which also introduces higher-rank analogues of geometric finiteness.

We generalize to a wider class of hyperbolic groups a construction by Misha Kapovich yielding convex cocompact representations into real hyperbolic space.

For quotients of the $n+1$-dimensional hyperbolic space by a convex co-compact group $\Gamma$, we obtain a formula relating the renormalized trace of the wave operator with the resonances of the Laplacian and some conformal invariants of the boundary, generalizing a formula of Guillope and Zworski in dimension 2. By writing this trace with the length spectrum, we prove precise asymptotics of the number of closed geodesics with an effective, exponentially small error term when the dimension of the limit set of $\Gamma$ is greater than $n/2$.

Relatively dominated representations give a common generalization of geometrically finiteness in rank one on the one hand, and the Anosov condition which serves as a higher-rank analogue of convex cocompactness on the other. This note proves three results about these representations. Firstly, we remove the quadratic gaps assumption involved in the original definition. Secondly, we give a characterization using eigenvalue gaps, providing a relative analogue of a result of Kassel and Potrie for Anosov representations. Thirdly, we formulate characterizations in terms of singular value or eigenvalue gaps combined with limit maps, in the spirit of Guéritaud et al. for Anosov representations, and use them to show that inclusion representations of certain groups playing weak ping-pong are relatively dominated.

We give a dynamical characterization of convex cocompact group actions on properly convex domains in projective space in the sense of Danciger–Guéritaud–Kassel: we show that convex cocompactness in is equivalent to an expansion property of the group about its limit set, occurring in different Grassmannians. As an application, we give a sufficient and necessary condition for convex cocompactness for groups that are hyperbolic relative to a collection of convex cocompact subgroups. We show that convex cocompactness in this situation is equivalent to the existence of an equivariant homeomorphism from the Bowditch boundary to the quotient of the limit set of the group by the limit sets of its peripheral subgroups.

In this paper, for a given Fuchsian group Γ, we prove an upper estimate for the Hausdorff dimension of the radial limit set in the visibility manifold. Further, if Γ is a convex cocompact group, we find the exact Hausdroff dimension of the limit set.

Given a convex representation \(\rho :\Gamma \rightarrow {{\mathrm{PGL}}}(d,\mathbb {R})\) of a convex cocompact group \(\Gamma \) of \({{\mathrm{Isom}}}_+\mathbb {H}^k,\) we find upper bounds for the quantity \(\alpha h_\rho ,\) where \(h_\rho \) is the entropy of \(\rho \) and \(\alpha \) is the Hölder exponent of the equivariant map \({{\partial }_{\infty }}\Gamma \rightarrow \mathbb {P}(\mathbb {R}^d).\) We also give rigidity statements when the upper bound is attained. This provides an analog of Thurston’s metric for convex cocompact groups of \({{\mathrm{Isom}}}_+\mathbb {H}^k.\) We then prove that if \(\rho :\pi _1\Sigma \rightarrow {{\mathrm{PSL}}}(d,\mathbb {R})\) is in the Hitchin component then \(\alpha h_\rho \le 2/(d-1)\) (where \(\alpha \) is the Hölder exponent of Labourie’s equivariant flag curve) with equality if and only if \(\rho \) is Fuchsian.

Limit sets and convex cocompact groups in higher rank symmetric spaces

In this paper, we consider convex co-compact subgroups of the projective linear group. We prove that such a group is relatively hyperbolic with respect to a collection of virtually Abelian subgroups of rank 2 if and only if each open face in the ideal boundary has dimension at most one. We also introduce the “coarse Hilbert dimension” of a subset of a convex set and use it to characterize when a naive convex co-compact subgroup is word hyperbolic or relatively hyperbolic with respect to a collection of virtually Abelian subgroups of rank 2.

Let X be a globally symmetric space of noncompact type, \( G = \textrm{Isom}^o (X) \) and \( \Gamma \subset G \) a discrete subgroup. Introducing an appropriate notion of Hausdorff measure on the geometric boundary \( \theta X \) of \( \theta X \), we prove that for regular boundary points \( \xi \in \theta X \) , the Hausdorff dimension of the radial limit set in \( G \cdot \xi \) is bounded above by the exponential growth rate of the number of orbit points close in direction to \( G \cdot \xi \subseteq \theta X \). Furthermore, for Zariski dense discrete groups Γ we construct Γ-invariant densities with support in every G-invariant subset of the limit set and study their properties. For a class of groups which generalises convex cocompact groups in the rank one setting, these densities allow to give a sharp estimate on the Hausdorff dimension of the radial limit set in each subset \( G \cdot \xi \subseteq \theta X \).