arXiv subjects

Pratyush Sarkar

Publications and source records attributed to Pratyush Sarkar.

14 recordsLinked to original sources

arXiv2026

Exponential prime orbit theorems for Anosov subgroups

Let $Γ$ be a Zariski dense Anosov subgroup of a connected semisimple real algebraic group -- these are higher rank analogues of convex cocompact subgroups. Let us measure the Jordan projections with any linear form which is positive on the limit cone of $Γ$. We prove a corresponding counting theorem with a power saving error term for the conjugacy classes of loxodromic elements in $Γ$. The proof is based on interpreting the Jordan projections as periods of a natural flow associated to $Γ$ and proving exponential mixing. We also prove the existence of a spectral gap for the Selberg zeta function.

math.DS↗

arXiv2026

Generalization of Selberg's $3/16$ theorem for geometrically finite thin subgroups of $\operatorname{SO}(n, 1)$

Let $Γ$ be a geometrically finite thin subgroup of an arithmetic lattice $Γ_0 < G := \operatorname{SO}(n, 1)$ and consider the congruence covers of $Γ\backslash G$. In the breakthrough work of Bourgain-Gamburd-Sarnak, the expansion machinery was used to establish a uniform spectral gap in the setting $(G, Γ_0) = (\operatorname{SL}_2(\mathbb{R}), \operatorname{SL}_2(\mathbb{Z}))$ when the critical exponent satisfies $δ_Γ> \frac{1}{2}$. The main applications are affine sieve for $Γ$-orbits and uniform resonance-free half-planes for the resolvent of the Laplacian. These results were generalized in subsequent works by Mohammadi-Oh, Oh-Winter, the author, and Edwards-Oh. Yet, the region $δ_Γ\in \bigl(\frac{1}{2}, n - 2\bigr]$ for $n \geq 3$ remains to be treated when there are cusps. The purpose of this paper is to fill in this gap in the literature. The difficulty lies in working with a countably infinite coding due to the presence of cusps. In particular, we incorporate new tools to prove the Zariski density and full trace field properties of the return trajectory subgroups.

math.DS↗

arXiv2026

A dynamical proof of non-arithmeticity of Jordan spectra

We give a dynamical proof of Benoist's non-arithmeticity theorem for Jordan spectra of Zariski dense subgroups of connected semisimple real algebraic groups. After passing to a Zariski dense Schottky subgroup, we use the coding of the limit set to realize Jordan projections as periods of a vector-valued Busemann return map for an expanding map on the Furstenberg boundary. The key step is to prove that a suitable two-branch asymptotic discrepancy is not locally constant on the limit set. We also show that the same criterion applies beyond Lie groups; in particular, it yields a direct density result for multiplier spectra of hyperbolic rational maps whose Julia set is not contained in a circle.

math.DS↗

arXiv2026

Effective equidistribution of translates of tori in arithmetic homogeneous spaces and applications

Let $Γ< G$ be an arithmetic lattice in a noncompact connected semisimple real algebraic group. For many such $G$ of rank at most $2$, in particular $G = \operatorname{SL}_3(\mathbb R)$, we prove effective equidistribution of large translates of tori in $G/Γ$. As an application, we obtain an asymptotic counting formula with a power saving error term for integral $3 \times 3$ matrices with a specified characteristic polynomial. These effectivize celebrated theorems of Eskin-Mozes-Shah.

math.DS↗

arXiv2026

Strong spectral gap for geometrically finite hyperbolic manifolds

Let $Γ< G := \operatorname{SO}(d+1, 1)$ for $d \geq 1$ be a Zariski dense, geometrically finite, discrete subgroup with critical exponent strictly greater than $d/2$. We show that $L^2(Γ\backslash G)$ admits a strong spectral gap, confirming a conjecture of Mohammadi and Oh. This extends the spherical spectral gap on $L^2(Γ\backslash \mathbb{H}^{d+1}) \cong L^2(Γ\backslash G/\operatorname{SO}(d+1))$, which follows by the works of Lax-Phillips, Patterson, and Sullivan by different methods. As a consequence, we establish rates of decay of matrix coefficients, and of exponential mixing of the frame flow, that are explicitly determined by the size of the strong spectral gap.

math.DS↗

arXiv2025

Congruence counting in Schottky and continued fractions semigroups of $\operatorname{SO}(n, 1)$

In this paper, the two settings we are concerned with are $Γ< \operatorname{SO}(n, 1)$ a Zariski dense Schottky semigroup and $Γ< \operatorname{SL}_2(\mathbb C)$ a Zariski dense continued fractions semigroup. In both settings, we prove a uniform asymptotic counting formula for the associated congruence subsemigroups, generalizing the work of Magee-Oh-Winter [arXiv:1601.03705] in $\operatorname{SL}_2(\mathbb R)$ to higher dimensions. Superficially, the proof requires two separate strategies: the expander machinery of Golsefidy-Varjú, based on the work of Bourgain-Gamburd-Sarnak, and Dolgopyat's method. However, there are several challenges in higher dimensions. Firstly, using the expander machinery requires a key input: the Zariski density and full trace field property of the return trajectory subgroups, newly introduced in [arXiv:2006.07787]. Secondly, we need to adapt Stoyanov's version of Dolgopyat's method to circumvent some technical issues while the main difficulty is to prove the key inputs: the local non-integrability condition (LNIC) and the non-concentration property (NCP).

math.NT↗

arXiv2025

Exponential mixing of frame flows for convex cocompact locally symmetric spaces

Let $G$ be a connected center-free simple real algebraic group of rank one and $Γ< G$ be a Zariski dense torsion-free convex cocompact subgroup. We prove that the frame flow on $Γ\backslash G$, i.e., the right translation action of a one-parameter subgroup $\{a_t\}_{t \in \mathbb R} < G$ of semisimple elements, is exponentially mixing with respect to the Bowen-Margulis-Sullivan measure. The key step is proving suitable generalizations of the local non-integrability condition and the non-concentration property which are essential for Dolgopyat's method. This generalizes the work of Sarkar-Winter for $G = \operatorname{SO}(n, 1)^\circ$ and also strengthens the mixing result of Winter in the convex cocompact case.

math.DS↗

arXiv2024

Finitary estimates for the distribution of lattice orbits in homogeneous spaces I: Riemannian metric

Let $H < G$ both be noncompact connected semisimple real algebraic groups where the former is maximal proper and $Γ< G$ be a lattice. Building on the work of Gorodnik-Weiss, we refine their techniques and obtain effective results. More precisely, we prove effective convergence of the distribution of dense $Γ$-orbits in $G/H$ to some limiting density on $G/H$ assuming effective equidistribution of regions of maximal horospherical orbits under one-parameter diagonal flows inside a dense $H$-orbit in $Γ\backslash G$. The significance of the effectivized argument is due to the recent effective equidistribution results of Lindenstrauss-Mohammadi-Wang for $Δ(\operatorname{SL}_2(\mathbb R)) < \operatorname{SL}_2(\mathbb R) \times \operatorname{SL}_2(\mathbb R)$ and $\operatorname{SL}_2(\mathbb R) < \operatorname{SL}_2(\mathbb C)$ and arithmetic lattices $Γ$, and future generalizations in that direction.

math.DS↗

arXiv2024

Uniform exponential mixing for congruence covers of convex cocompact hyperbolic manifolds

Let $Γ$ be a Zariski dense convex cocompact subgroup contained in an arithmetic lattice of $\operatorname{SO}(n, 1)^{\circ}$. We prove uniform exponential mixing of the geodesic flow for congruence covers of the hyperbolic manifold $Γ\backslash \mathbb{H}^n$ avoiding finitely many prime ideals. This extends the work of Oh-Winter who proved the result for the $n = 2$ case. Following their approach, we use Dolgopyat's method for the proof of exponential mixing of the geodesic flow. We do this uniformly over congruence covers by establishing uniform spectral bounds for the congruence transfer operators associated to the geodesic flow. This requires another key ingredient which is the expander machinery due to Bourgain-Gamburd-Sarnak extended by Golsefidy-Varjú.

math.DS↗

arXiv2024

Exponential mixing of frame flows for convex cocompact hyperbolic manifolds

The aim of this paper is to establish exponential mixing of frame flows for convex cocompact hyperbolic manifolds of arbitrary dimension with respect to the Bowen-Margulis-Sullivan measure. Some immediate applications include an asymptotic formula for matrix coefficients with an exponential error term as well as the exponential equidistribution of holonomy of closed geodesics. The main technical result is a spectral bound on transfer operators twisted by holonomy, which we obtain by building on Dolgopyat's method.

math.DS↗

arXiv2022

Local mixing of one-parameter diagonal flows on Anosov homogeneous spaces

Let $G$ be a connected semisimple real algebraic group and $Γ< G$ be a Zariski dense Anosov subgroup with respect to a minimal parabolic subgroup. We prove local mixing of the one-parameter diagonal flow $\{\exp(t\mathsf{v}) : t \in \mathbb R\}$ on $Γ\backslash G$ for any interior direction $\mathsf{v}$ of the limit cone of $Γ$ with respect to the Bowen--Margulis--Sullivan measure associated to $\mathsf{v}$. More generally, we allow a class of deviations to this flow along a direction $\mathsf{u}$ in some fixed subspace transverse to $\mathsf{v}$. We also obtain a uniform bound for the correlation function which decays exponentially in $\|\mathsf{u}\|^2$. The precise form of the result is required for several applications such as the asymptotic formula for the decay of matrix coefficients in $L^2(Γ\backslash G)$ proved by Edwards--Lee--Oh.

math.DS↗

arXiv2022

Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps

Let $Γ$ be a geometrically finite discrete subgroup in $\operatorname{SO}(d+1,1)^{\circ}$ with parabolic elements. We establish exponential mixing of the geodesic flow on the unit tangent bundle $\operatorname{T}^1(Γ\backslash \mathbb{H}^{d+1})$ with respect to the Bowen-Margulis-Sullivan measure, which is the unique probability measure on $\operatorname{T}^1(Γ\backslash \mathbb{H}^{d+1})$ with maximal entropy. As an application, we obtain a resonance free region for the resolvent of the Laplacian on $Γ\backslash \mathbb{H}^{d+1}$. Our approach is to construct a coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator.

math.DS↗

arXiv2020

Generalization of Selberg's $3/16$ theorem for convex cocompact thin subgroups of $\operatorname{SO}(n, 1)$

Let $Γ$ be a convex cocompact thin subgroup of an arithmetic lattice in $\operatorname{SO}(n, 1)$. We generalize Selberg's $\frac{3}{16}$ theorem in this setting, i.e., we prove uniform exponential mixing of the frame flow and obtain a uniform resonance-free half plane for the congruence covers of the hyperbolic manifold $Γ\backslash \mathbb H^n$. This extends the work of Oh-Winter who established the $n = 2$ case. The theorem follows from uniform spectral bounds for the congruence transfer operators with holonomy. We employ Sarkar-Winter's frame flow version of Dolgopyat's method uniformly over the congruence covers as well as Golsefidy-Varjú's generalization of Bourgain-Gamburd-Sarnak's expansion machinery by using the properties that the return trajectory subgroups are Zariski dense and have trace fields which coincide with that of $Γ$. These properties follow by proving that the return trajectory subgroups have finite index in $Γ$.

math.DS↗

arXiv2019

Bifurcating entanglement-renormalization group flows of fracton stabilizer models

We investigate the entanglement-renormalization group flows of translation-invariant topological stabilizer models in three dimensions. Fracton models are observed to bifurcate under entanglement renormalization, generically returning at least one copy of the original model. Based on this behavior we formulate the notion of bifurcated equivalence for fracton phases, generalizing foliated fracton equivalence. The notion of quotient superselection sectors is also generalized accordingly. We calculate bifurcating entanglement-renormalization group flows for a wide range of examples and, based on those results, propose conjectures regarding the classification of translation-invariant topological stabilizer models in three dimensions.

cond-mat.str-el↗