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Sebastian Hurtado

Publications and source records attributed to Sebastian Hurtado.

17 recordsLinked to original sources
arXiv2026

On some aspects of discrete groups acting ergodically on the boundary

We show that if $G$ is a real semisimple Lie group and $\Gamma 2$. The examples arise from lattices $\Gamma<H= SO(n,1)$ and their deformations in $G$. Perhaps more importantly, we describe a new approach to studying deformations of such lattices by relating them to deformations of the smooth right-translation action of $SO(n-1,1)$ on $\Gamma\backslash H$. This correspondence allows us to apply recent results of DeWitt and Dolgopyat on smooth group actions and to give examples where the right-translation action of $SO(n-1,1)$ on $\Gamma\backslash H$ can fail to be $C^0$-locally rigid.

math.DS↗
arXiv2025

Irreducible groups and ergodicity in the boundary

We show that if $G$ is a real semi-simple Lie group, and $\Gamma$ is a discrete subgroup of $G$ containing a subgroup $\Sigma$ acting ergodically (in a strong sense) on the Furstenberg boundary of $G$, then $\Gamma$ is not isomorphic to a free product of $\Sigma$ with $\mathbb{Z}$. Moreover, if $\Sigma$ has algebraic entries, then $\Gamma$ has algebraic entries as well. As a consequence, we show that if all irreducible discrete subgroups of ${\rm SL}_2(\mathbb{R}) \times {\rm SL}_2(\mathbb{R})$ act ergodically on $\mathbb{S}^1\times \mathbb{S}^1 $, such groups cannot be free groups (or even Gromov hyperbolic). In the appendix, we discuss a connection between the existence of discrete irreducible groups and diophantine properties of Lie groups.

math.GR↗
arXiv2025

A strong height gap theorem for $PGL_2$

The height gap theorem states that the finite subsets $F$ of matrices generating non-virtually solvable groups have normalized height $\widehat{h}(F)$ bounded below by a constant. It was first proved by Breuillard and another proof was given later by Chen, Hurtado and Lee. In this paper we show that when the set $F$ is contained in a maximal arithmetic subgroup $\Gamma$ of $G = PGL_2(\mathbb{R})^a \times PGL_2(\mathbb{C})^b$, $a+b \ge 1$, the height bound for the case when $F$ generates a Zariski dense subgroup of $G$ over $\mathbb{R}$ is proportional to $\log(covol(\Gamma))$, the function of the covolume of $\Gamma$. This result strengthens the theorem for the lattices of large covolume and has various applications including a strong version of the arithmetic Margulis lemma for $PGL_2(\mathbb{R})^a \times PGL_2(\mathbb{C})^b$.

math.GR↗
arXiv2024

Remarks on discrete subgroups with full limit sets in higher rank Lie groups

We show that real semi-simple Lie groups of higher rank contain (infinitely generated) discrete subgroups with full limit sets in the corresponding Furstenberg boundaries. Additionally, we provide criteria under which discrete subgroups of $G = \operatorname{SL}(3,\mathbb{R})$ must have a full limit set in the Furstenberg boundary of $G$. In the appendix, we show the the existence of Zariski-dense discrete subgroups $\Gamma$ of $\operatorname{SL}(n,\mathbb{R})$, where $n\ge 3$, such that the Jordan projection of some loxodromic element $\gamma \in\Gamma$ lies on the boundary of the limit cone of $\Gamma$.

math.GT↗
arXiv2022

A new proof of finiteness of maximal arithmetic reflection groups

We give a new proof of the finiteness of maximal arithmetic reflection groups. Our proof is novel in that it makes no use of trace formulas or other tools from the theory of automorphic forms and instead relies on the arithmetic Margulis lemma of Fraczyk, Hurtado and Raimbault.

math.GT↗
arXiv2022

Topological complexity of arithmetic locally symmetric spaces

We prove that any arithmetic locally symmetric space is homotopy equivalent to a simplicial complex where the number of simplices is bounded linearly in the volume of the space. This settles a well-known conjecture of Gelander. The main technical ingredient, which is of independent interest, is a strengthened version of the Margulis' collar lemma for arithmetic locally symmetric spaces based on the height gap theorem of Breuillard, in which the Margulis constant is made linear in the degree of the trace field of the lattice.

math.NT↗
arXiv2021

A height gap in $GL_d(\overline{\mathbb{Q}})$ and almost laws

E. Breuillard showed that finite subsets $F$ of matrices in $GL_d(\overline{\mathbb{Q}})$ generating non-virtually solvable groups have normalized height $\widehat{h}(F) \ge ε_d$, for some positive $ε_d >0$. The normalized height $\widehat{h}(F)$ is a measure of the arithmetic size of $F$ and this result can be thought of as a non-abelian analog of Lehmer's Mahler measure problem. We give a new shorter proof of this result. Our key idea relies on the existence of particular word maps in compact Lie groups (known as almost laws) whose image lies close to the identity element.

math.GR↗
arXiv2021

Zimmer's conjecture for non-uniform lattices: escape of mass and growth of cocycles

We establish finiteness of low-dimensional actions of lattices in higher-rank semisimple Lie groups and establish Zimmer's conjecture for many such groups. This builds on previous work of the authors handling the case of actions by cocompact lattices and of actions by $\Sl(n,\Z)$. While the results are not sharp in all cases, they do dramatically improve all known results. The key difficulty overcome in this paper concerns escape of mass when taking limits of sequences of measures. Due to a need to control Lyapunov exponents for unbounded cocycles when taking such limits, quantitative controls on the concentration of mass at infinity are need and novel techniques are introduced to avoid ``escape of Lyapunov exponent."

math.DS↗
arXiv2020

Global Rigidity of Some Abelian-by-Cyclic group actions on $\T^2$

For groups of diffeomorphisms of $\T^2$ containing an Anosov diffeomorphism, we give a complete classification for polycyclic Abelian-by-Cyclic group actions on $\T^2$ up to both topological conjugacy and smooth conjugacy under mild assumptions. Along the way, we also prove a Tits alternative type theorem for some groups of diffeomorphisms of $\T^2$.

math.DS↗
arXiv2020

Zimmer's conjecture: Subexponential growth, measure rigidity, and strong property (T)

We prove several cases of Zimmer's conjecture for actions of higher-rank cocompact lattices on low dimensional manifolds. For example, if $Γ$ is a cocompact lattice in $\mathrm{Sl}(n, \mathbb R)$, $M$ is a compact manifold, and $ω$ a volume form on $M$ we show that any homomorphism $ρ\colon Γ\rightarrow \mathrm{Diff}(M)$ has finite image if the dimension of $M$ is less than $n-1$ and that any homomorphism $ρ\colon Γ\rightarrow \mathrm{Diff}(M,ω)$ has finite image if the dimension of $M$ is less than $n$. The key step in the proof is to show any such action has uniform subexponential growth of derivatives. This is established using ideas from the smooth ergodic theory of higher-rank abelian groups, structure theory of semisimple groups and results from homogeneous dynamics. Having established uniform subexponential growth of derivatives, we apply Lafforgue's strong property (T) to establish the existence of an invariant Riemannian metric.

math.DS↗
arXiv2020

Zimmer's conjecture for actions of $\mathrm{SL}(m,\mathbb{Z})$

We prove Zimmer's conjecture for $C^2$ actions by finite-index subgroups of $\mathrm{SL}(m,\mathbb{Z})$ provided $m>3$. The method utilizes many ingredients from our earlier proof of the conjecture for actions by cocompact lattices in $\mathrm{SL}(m,\mathbb{R})$ but new ideas are needed to overcome the lack of compactness of the space $(G \times M)/Γ$ (admitting the induced $G$-action). Non-compactness allows both measures and Lyapunov exponents to escape to infinity under averaging and a number of algebraic, geometric, and dynamical tools are used control this escape. New ideas are provided by the work of Lubotzky, Mozes, and Raghunathan on the structure of nonuniform lattices and, in particular, of $\mathrm{SL}(m,\mathbb{Z})$ providing a geometric decomposition of the cusp into rank one directions, whose geometry is more easily controlled. The proof also makes use of a precise quantitative form of non-divergence of unipotent orbits by Kleinbock and Margulis, and an extension by de la Salle of strong property (T) to representations of nonuniform lattices.

math.DS↗
arXiv2019

The Burnside problem for $\text{Diff}_{\text{Vol}}(\mathbb{S}^2)$

Let $S$ be a closed surface and $\text{Diff}_{\text{Vol}}(S)$ be the group of volume preserving diffeomorphisms of $S$. A finitely generated group $G$ is periodic of bounded exponent if there exists $k \in \mathbb{N}$ such that every element of $G$ has order at most $k$. We show that every periodic group of bounded exponent $G \subset \text{Diff}_{\text{Vol}}(S)$ is a finite group.

math.DS↗
arXiv2017

Examples of diffeomorphism group cocycles with no periodic approximation

We construct a finitely generated subgroup of $\text{Diff}^{\infty}(\mathbb{S}^3 \times \mathbb{S}^1)$ where every element is conjugate to an isometry but such that the group action itself is far from isometric (the group has "exponential growth of derivatives"). As a corollary, one obtains a locally constant $\text{Diff}^{\infty}(\mathbb{S}^3 \times \mathbb{S}^1)$ valued cocycle over a hyperbolic dynamical system which has elliptic behavior over its periodic orbits but which preserves a measure with non-zero top Fiber Lyapunov exponent. Additionally, we provide new examples of Banach cocycles not satisfying the periodic approximation property as first shown by Kalinin-Sadovskaya.

math.DS↗
arXiv2016

Distortion and Tits alternative in smooth mapping class groups

In this article, we study the smooth mapping class group of a surface S relative to a given Cantor set, that is the group of isotopy classes of orientation-preserving smooth diffeomorphisms of S which preserve this Cantor set. When the Cantor set is the standard ternary Cantor set, we prove that the subgroup consisting of diffeomorphisms which are isotopic to the identity on S does not contain any distorted elements. Moreover, we prove a weak Tits alternative for these groups.

math.DS↗
arXiv2016

Continuity of discrete homomorphisms of diffeomorphism groups

Let $M$ and $N$ be two closed $C^{\infty}$ manifolds and let $\text{Diff}_c(M)$ denote the group of $C^{\infty}$ diffeomorphisms isotopic to the identity. We prove that any (discrete) group homomorphism between $\text{Diff}_c(M)$ and $\text{Diff}_c(N)$ is continuous. We also show that a non-trivial group homomorphism $Φ: \text{Diff}_c(M) \to \text{Diff}_c(N)$ implies that $\dim(M) \leq \dim(N)$ and give a classification of such homomorphisms when $\dim(M) = \dim(N)$.

math.GT↗