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Tengren Zhang

Publications and source records attributed to Tengren Zhang.

At least 19 recordsLinked to original sources
arXiv2026

Geometric finiteness in paracomplex hyperbolic spaces

We develop a framework for studying discrete subgroups of $\mathsf{PGL}(d+1,\mathbb{R})$ via the paracomplex hyperbolic space $\mathbb{H}_\tau^d$, a rank-$1$ pseudo-Riemannian symmetric space. We characterize projective transverse, relatively Anosov, and Anosov subgroups in terms of properly discontinuous, geometrically finite, and convex-cocompact actions respectively on their weak hulls, which are canonical flow spaces in the spacelike unit tangent bundle of $\mathbb{H}_\tau^d$. A key ingredient is the construction of a Busemann-type horofunction on the spacelike unit tangent bundle with the properties needed to describe cuspidal geometry. We further prove for relatively Anosov subgroups that the geodesic flows on their weak hulls are uniformly hyperbolic, giving a relative analogue of the Axiom A property.

math.DG↗
arXiv2026

Orbital counting for relatively Anosov groups

We obtain orbital counting results for relatively Anosov groups with respect to linear functionals with finite critical exponent. Our counting results follow from an equidistribution result and rely crucially on previous equidistribution results obtained in our proof of counting results for periods. Our results generalize earlier work of Sambarino in the setting of Anosov groups.

math.DS↗
arXiv2026

Positive representations over real closed fields

We develop the theory of $\Theta$-positive representations from general Fuchsian groups to linear groups over real closed fields. Our definition, which does not assume the boundary map to be continuous, encompasses many generalizations of positive or Anosov representations that have been considered in the literature.

math.GT↗
arXiv2025

A rigidity theorem for complex Kleinian groups

Farre, Pozzetti and Viaggi proved that any (d-k)-hyperconvex subgroup of PSL(d,C) is virtually isomorphic to a convex cocompact Kleinian group and that its k-th simple root critical exponent is at most 2. We show that a (d-k)-hyperconvex subgroup is isomorphic to a uniform lattice in PSL(2,C) if and only if its k-th simple root critical exponent is exactly 2. Furthermore, we show that if a strongly irreducible (d-k)-hyperconvex subgroup has k-th simple root critical exponent 2, then it is the image of a uniform lattice in PSL(2, C) by an irreducible representation of PSL(2, C) into PSL(d, C).

math.DG↗
arXiv2025

Topology of the space of $d$-pleated surfaces

Given a maximal geodesic lamination $\lambda$ on a closed oriented surface $S$ of genus $g$, the space of $d$-pleated surfaces with pleating locus $\lambda$ is an open subset of $\mathrm{Hom}(\pi_1(S),\mathsf{PGL}_d(\mathbb{C}))$ obtained by applying generalized bending along $\lambda$ to Hitchin representations. When $d=2$, one recovers abstract pleated surfaces in $\mathbb{H}^3$. In this paper, we study the topology of the space $\mathfrak{R}(\lambda,d)$ of conjugacy classes of $d$-pleated surfaces with pleating locus $\lambda$. Firstly, we prove that $\mathfrak{R}(\lambda,d)$ is real-analytically diffeomorphic to $\mathbb{R}^{(d^2-1)(2g-2)}\times(\mathbb{R}/2\pi\mathbb{Z})^{(d^2-1)(2g-2)}\times \mathbb{Z}_d$, where $\mathbb{Z}_d$ denotes the finite cyclic group of order $d$. Furthermore, we show that each connected component of the space of conjugacy classes in $\mathrm{Hom}(\pi_1(S),\mathsf{PGL}_d(\mathbb{C}))$ contains exactly one component of $\mathfrak{R}(\lambda,d)$.

math.GT↗
arXiv2024

Regularity of limit sets of Anosov representations

In this paper we establish necessary and sufficient conditions for the limit set of a projective Anosov representation to be a differentiable submanifold of projective space with Holder continuous derivatives. We also calculate the optimal value of the Holder constant in terms of the eigenvalue data of the Anosov representation.

math.DG↗
arXiv2024

Patterson-Sullivan measures for transverse subgroups

We study Patterson-Sullivan measures for a class of discrete subgroups of higher rank semisimple Lie groups, called transverse groups, whose limit set is well-defined and transverse in a partial flag variety. This class of groups includes both Anosov and relatively Anosov groups, as well as all discrete subgroups of rank one Lie groups. We prove an analogue of the Hopf-Tsuji-Sullivan dichotomy and then use this dichotomy to prove a variant of Burger's Manhattan curve theorem. We also use the Patterson-Sullivan measures to obtain conditions for when a subgroup has critical exponent strictly less than the original transverse group. These gap results are new even for Anosov groups.

math.GR↗
arXiv2023

The Goldman symplectic form on the PGL(V)-Hitchin component

This article is the second of a pair of articles about the Goldman symplectic form on the PGL(V)-Hitchin component of a closed, connected, oriented, hyperbolic surface S. We show that any ideal triangulation on S and any compatible bridge system determine a symplectic trivialization of the tangent bundle to the PGL(V)-Hitchin component of S. Using this, we prove that a large class of vector fields defined in the companion paper [SWZ20] are Hamiltonian. This is then used to prove that the explicit global coordinate system defined in the companion paper [SWZ20] is a global Darboux coordinate system for the PGL(V)-Hitchin component.

math.DG↗
arXiv2023

Patterson-Sullivan measures for relatively Anosov groups

We establish existence, uniqueness and ergodicity results for Patterson-Sullivan measures for relatively Anosov groups. As applications we obtain an entropy gap theorem and a strict concavity result for entropies associated to linear functionals.

math.DS↗
arXiv2023

Cusped Borel Anosov representations with positivity

We show that if a cusped Borel Anosov representation from a lattice $Γ\subset \mathsf{PGL}_2(\mathbb{R})$ to $\mathsf{PGL}_d(\mathbb{R})$ contains a unipotent element with a single Jordan block in its image, then it is necessarily a (cusped) Hitchin representation. We also show that the amalgamation of a Hitchin representation with a cusped Borel Anosov representation that is not Hitchin is never cusped Borel Anosov.

math.DG↗
arXiv2023

$d$-pleated surfaces and their shear-bend coordinates

In this article, we single out representations of surface groups into $\mathsf{PSL}_d(\mathbb{C})$ which generalize the well-studied family of pleated surfaces into $\mathsf{PSL}_2(\mathbb{C})$. Our representations arise as sufficiently generic $\lambda$-Borel Anosov representations, which are representations that are Borel Anosov with respect to a maximal geodesic lamination $\lambda$. For fixed $\lambda$ and $d$, we provide a holomorphic parametrization of the space $\mathcal{R}(\lambda,d)$ of $(\lambda,d)$-pleated surfaces which extends both work of Bonahon for pleated surfaces and Bonahon and Dreyer for Hitchin representations.

math.GT↗
arXiv2022

Weakly positive and directed Anosov representations

Given a finitely generated group $Γ$, a directed graph $Λ$, and a map $R:Λ\toΓ$, we introduce the notion of an $(R,Λ)$-directed Anosov representation. This is a weakening of the notion of Anosov representations. Our main theorem gives a procedure to construct $(R,Λ)$-directed Anosov representations using Fock-Goncharov positivity. As an application of our main theorem, we construct large families of primitive stable representations from $F_2$ to $\mathrm{PGL}(V)$, including non-discrete and non-faithful examples.

math.GT↗
arXiv2022

Cusped Hitchin representations and Anosov representations of geometrically finite Fuchsian groups

We develop a theory of Anosov representation of geometrically finite Fuchsian groups in SL(d,R) and show that cusped Hitchin representations are Borel Anosov in this sense. We establish analogues of many properties of traditional Anosov representations. In particular, we show that our Anosov representations are stable under type-preserving deformations and that their limit maps vary analytically. We also observe that our Anosov representations fit into the previous frameworks of relatively Anosov and relatively dominated representations developed by Kapovich-Leeb and Zhu.

math.DG↗
arXiv2022

Entropy rigidity for cusped Hitchin representations

We establish an entropy rigidity theorem for Hitchin representations of all geometrically finite Fuchsian groups which generalizes a theorem of Potrie and Sambarino for Hitchin representations of closed surface groups. In the process, we introduce the class of (1,1,2)-hypertransverse groups and show for such a group that the Hausdorff dimension of its conical limit set agrees with its (first) simple root entropy, providing a common generalization of results of Bishop and Jones, for Kleinian groups, and Pozzetti, Sambarino and Wienhard, for Anosov groups. We also introduce the theory of transverse representations of projectively visible groups as a tool for studying discrete subgroups of linear groups which are not necessarily Anosov or relatively Anosov.

math.GR↗
arXiv2021

Coordinates on the augmented moduli space of convex RP^2 structures

Let S be an orientable, finite type surface with negative Euler characteristic. The augmented moduli space of convex real projective structures on S was first defined and topologized by the first author. In this article, we give an explicit description of this topology using explicit coordinates. More precisely, given every point in this augmented moduli space, we find explicit continuous coordinates on the quotient of a suitable open neighborhood about this point by a suitable subgroup of the mapping class group of S. Using this, we give a simpler proof of the fact that the augmented moduli space of convex real projective structures on S is homeomorphic to the orbifold vector bundle of regular cubic differentials over the Deligne-Mumford compactification of the moduli space of Riemann surfaces homeomorphic to S.

math.DG↗
arXiv2019

Flows on the PGL(V)-Hitchin component

In this article we define new flows on the Hitchin components for PGL(V). Special examples of these flows are associated to simple closed curves on the surface and give generalized twist flows. Other examples, so called eruption flows, are associated to pair of pants in S and capture new phenomena which are not present in the case when n = 2. Using these flows, we construct a global coordinate system on the Hitchin component. In a companion paper to this article two of the authors develop new tools to compute the Goldman symplectic form on the Hitchin component, and prove that this global coordinate system is a Darboux coordinate system.

math.DG↗
arXiv2019

Positively ratioed representations

Let S be a closed orientable surface of genus at least 2 and let G be a semisimple real algebraic group of non-compact type. We consider a class of representations from the fundamental group of S to G called positively ratioed representations. These are Anosov representations with the additional condition that certain associated cross ratios satisfy a positivity property. Examples of such representations include Hitchin representations and maximal representations. Using geodesic currents, we show that the corresponding length functions for these positively ratioed representations are well-behaved. In particular, we prove a systolic inequality that holds for all such positively ratioed representations.

math.GT↗