arXiv subjects

Stephen Cantrell

Publications and source records attributed to Stephen Cantrell.

At least 19 recordsLinked to original sources

arXiv2026

A geometric correspondence for reparameterizations of geodesic flows

For any non-elementary, torsion-free hyperbolic group, we provide a correspondence between the left-invariant Gromov-hyperbolic metrics on the group that are quasi-isometric to a word metric, and continuous reparameterizations of the associated Mineyev's flow space. From this correspondence, we produce the first examples of continuous reparameterizations of geodesic flows on negatively curved manifolds with all periodic orbits having integer lengths. For surface and free groups, this also yields isometric actions on Gromov-hyperbolic spaces on which loxodromic elements are precisely the non-simple elements. Key ingredients in our proof are an analysis of the geometry of Mineyev's flow space (such as the metric-Anosov property recently proven by Dilsavor), and the density of Green metrics in the moduli space of (symmetric) metrics on the group.

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arXiv2025

Multidimensional statistics for finite orbits of generalised continued fractions

We statistically compare the relationships between frequencies of digits in continued fraction expansions of typical rational points in the unit interval and higher dimensional generalisations. This takes the form of a Large Deviation and Central Limit Theorem, including multidimensional results for random vectors. These results apply to classical multidimensional continued fraction transformations including Brun's algorithm and the Jacobi--Perron algorithm, and more generally for maps satisfying mild contraction hypothesis on the inverse branches. We prove in particular that the finite trajectories capture the generic ergodic behaviour of infinite trajectories.

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arXiv2025

Sparse spectrally rigid sets for negatively curved manifolds

Suppose that $(M,\mathfrak{g})$ is a compact Riemannian manifold with strictly negative sectional curvatures. A subset of conjugacy classes $E \subset \text{conj}(π_1(M))$ is called spectrally rigid if when two negatively curved Riemannian metrics $\mathfrak{g}_1, \mathfrak{g}_2$ on $M$ have the same marked length spectrum on $E$, then their marked length spectra coincide everywhere. In this work we show that there are arbitrarily sparse spectrally rigid sets and that they exist, in some sense, in every direction in $π_1(M)$.

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arXiv2025

The Manhattan curve, ergodic theory of topological flows and rigidity

For every non-elementary hyperbolic group, we introduce the Manhattan curve associated to any pair of left-invariant hyperbolic metrics which are quasi-isometric to a word metric. It is convex; we show that it is continuously differentiable and moreover is a straight line if and only if the corresponding two metrics are roughly similar, i.e., they are within bounded distance after multiplying by a positive constant. Further, we prove that the Manhattan curve associated to two strongly hyperbolic metrics is twice continuously differentiable. The proof is based on the ergodic theory of topological flows associated to general hyperbolic groups and analyzing the multifractal structure of Patterson-Sullivan measures. We exhibit some explicit examples including a hyperbolic triangle group and compute the exact value of the mean distortion for pairs of word metrics.

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arXiv2025

Counting statistics for geodesics on flat surfaces

We study counting limit laws that compare length functions on infinite graphs. We then apply these results to flat surfaces to obtain a statistical comparison between the geometric length and the number of singularities visited by geodesic paths.

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arXiv2025

Mixing of the Mineyev flow, orbital counting and Poincaré series for strongly hyperbolic metrics

We obtain orbital counting results for the class of strongly hyperbolic metrics on hyperbolic groups. To achieve this we combine ergodic theoretic techniques involving the Mineyev topological flow and symbolic dynamics. Our results apply to the Green metric associated to an admissible, finitely supported, symmetric random walk and to the Mineyev hat metric. We also describe the domain of analyticity for the Poincaré series associated to these metrics, prove mixing results for the Mineyev topological flow and obtain correlation asymptotics for pairs of metrics.

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arXiv2024

The joint translation spectrum and Manhattan manifolds

We define and study geometric versions of the Benoist limit cone and matrix joint spectrum, which we call the translation cone and the joint translation spectrum, respectively. These new notions allow us to generalize the study of embeddings into products of rank-one simple Lie groups and to compare group actions on different metric spaces, quasi-morphisms, Anosov representations and many other natural objects of study. We identify the joint translation spectrum with the image of the gradient function of a corresponding Manhattan manifold: a higher dimensional version of the well known and studied Manhattan curve. As a consequence we deduce many properties of the spectrum. For example we show that it is given by the closure of the set of all possible drift vectors associated to finitely supported, symmetric, admissible random walks on the associated group.

math.GR↗

arXiv2024

Approximate marked length spectrum rigidity in coarse geometry

We compare the marked length spectra of isometric actions of groups with non-positively curved features. Inspired by the recent works of Butt we study approximate versions of marked length spectrum rigidity. We show that for pairs of metrics, the supremum of the quotient of their marked length spectra is approximately determined by their marked length spectra restricted to an appropriate finite set of conjugacy classes. Applying this to fundamental groups of closed negatively curved Riemannian manifolds allows us to refine Butt's result. Our results however apply in greater generality and do not require the acting group to be hyperbolic. For example we are able to compare the marked length spectra associated to mapping class groups acting on their Cayley graphs or on the curve graph.

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arXiv2024

Central limit theorems for Green metrics on hyperbolic groups

Suppose we have two finitely supported, admissible, probability measures on a hyperbolic group $Γ$. In this article we prove that the corresponding two Green metrics satisfy a counting central limit theorem when we order the elements of $Γ$ according to one of the metrics. Our results also apply to various other metrics including length functions associated to Anosov representations and to group actions on hyperbolic metric spaces.

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arXiv2024

Marked length spectrum rigidity from rigidity on subsets

We introduce a new method for studying length spectrum rigidity problems based on a combination of ideas from dynamical systems and geometric group theory. This allows us to compare the marked length spectrum of metrics and distance-like functions coming from various geometric origins. Using our new perspective, we provide concise proofs of well-known length spectrum rigidity results and are able to extend classical results to a variety of new settings. Our methods rely on studying Manhattan curves and a coarse geometric analogue of Teichmüller space equipped with a symmetrized version of the Thurston metric.

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arXiv2024

Manhattan geodesics and the boundary of the space of metric structures on hyperbolic groups

For any non-elementary hyperbolic group $Γ$, we find an outer automorphism invariant geodesic bicombing for the space of metric structures on $Γ$ equipped with a symmetrized version of the Thurston metric on Techimüller space. We construct and study a boundary for this space and show that it contains many well-known pseudo metrics including those coming from actions on $\text{CAT}(0)$ cube complexes, real trees and coned-off Cayley graphs. As corollaries we deduce length spectrum rigidity results, regularity results for Manhattan curves, optimal growth rate results for Anosov representations and results regarding continuous extensions of translation distance functions to the space of geodesic currents. Using our results for geodesic currents we settle a conjecture of Bonahon in the negative.

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arXiv2024

Invariant measures of the topological flow and measures at infinity on hyperbolic groups

We show that for every non-elementary hyperbolic group, an associated topological flow space admits a coding based on a transitive subshift of finite type. Applications include regularity results for Manhattan curves, the uniqueness of measures of maximal Hausdorff dimension with potentials, and the real analyticity of intersection numbers for families of dominated representation, thus providing a direct proof of a result established by Bridgeman, Canary, Labourie and Sambarino in 2015.

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arXiv2023

Rigidity phenomena and the statistical properties of group actions on $\text{CAT}(0)$ cube complexes

We compare the marked length spectra of some pairs of proper and cocompact cubical actions of a non-virtually cyclic group on $\text{CAT}(0)$ cube complexes. The cubulations are required to be virtually co-special, have the same sets of convex-cocompact subgroups, and admit a contracting element. There are many groups for which these conditions are always fulfilled for any pair of cubulations, including non-elementary cubulable hyperbolic groups, many cubulable relatively hyperbolic groups, and many right-angled Artin and Coxeter groups. For these pairs of cubulations, we study the Manhattan curve associated to their combinatorial metrics. We prove that this curve is analytic and convex, and a straight line if and only if the marked length spectra are homothetic. The same result holds if we consider invariant combinatorial metrics in which the lengths of the edges are not necessarily one. In addition, for their standard combinatorial metrics, we prove a large deviation theorem with shrinking intervals for their marked length spectra. We deduce the same result for pairs of word metrics on hyperbolic groups. The main tool is the construction of a finite-state automaton that simultaneously encodes the marked length spectra of both cubulations in a coherent way, in analogy with results about (bi)combable functions on hyperbolic groups by Calegari and Fujiwara. The existence of this automaton allows us to apply the machinery of thermodynamic formalism for suspension flows over subshifts of finite type, from which we deduce our results.

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arXiv2022

Counting and boundary limit theorems for representations of Gromov-hyperbolic groups

Given a Gromov-hyperbolic group $G$ endowed with a finite symmetric generating set, we study the statistics of counting measures on the spheres of the associated Cayley graph under linear representations of $G$. More generally, we obtain a weak law of large numbers for subadditive functions, echoing the classical Fekete lemma. For strongly irreducible and proximal representations, we prove a counting central limit theorem with a Berry--Esseen type error rate and exponential large deviation estimates. Moreover, in the same setting, we show convergence of interpolated normalized matrix norms along geodesic rays to Brownian motion and a functional law of iterated logarithm, paralleling the analogous results in the theory of random matrix products. Our counting large deviation estimates provide a positive answer to a question of Kaimanovich--Kapovich--Schupp. In most cases, our counting limit theorems will be obtained from stronger almost sure limit laws for Patterson--Sullivan measures on the boundary of the group.

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arXiv2020

Statistical limit laws for hyperbolic groups

Using techniques from ergodic theory and symbolic dynamics, we derive statistical limit laws for real valued functions on hyperbolic groups. In particular, our results apply to convex cocompact group actions on $\text{CAT}(-1)$ spaces, and provide a precise statistical comparison between word length and displacement. After generalising our methods to the multidimensional setting, we prove that the abelianisation map satisfies a non-degenerate multidimensional central limit theorem. We also obtain local limit theorems for group homomorphisms and for the displacement function associated to certain actions.

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arXiv2020

Typical behaviour along geodesic rays in hyperbolic groups

In this note we study the limiting behaviour of real valued functions on hyperbolic groups as we travel along typical geodesic rays in the Gromov boundary of the group. Our results apply to group homomorphisms, certain quasimorphisms and to the displacement functions associated to convex cocompact group actions on CAT$(-1)$ metric spaces.

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arXiv2020

Comparison theorems for closed geodesics on negatively curved surfaces

In this note we present new asymptotic estimates comparing the word length and geodesic length of closed geodesics on surfaces with (variable) negative sectional curvatures. In particular, we provide an averaged comparison of these two important quantities and obtain precise statistical results, including a central limit theorem and a local limit theorem. Further, as a corollary we also improve an asymptotic formula of R. Sharp and the second author. Finally, we relate our results to recent work of Gekhtman, Taylor and Tiozzo.

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arXiv2019

Relative growth in hyperbolic groups

In this note we obtain estimates on the relative growth of normal subgroups of non-elementary hyperbolic groups, particularly those with free abelian quotient. As a corollary, we deduce that the associated relative growth series fail to be rational.

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