Thurston's Legacy at the Interface of Surface Dynamics, Train Tracks, and Arithmetic

Ahmad Rafiqi; Ayberk Zeytin

Home/Archive/arXiv

arXiv · 2610.13347

Thurston's Legacy at the Interface of Surface Dynamics, Train Tracks, and Arithmetic

Ahmad Rafiqi·Ayberk Zeytin

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this chapter, we discuss the interaction between low-dimensional dynamics, symbolic dynamics, and geometric topology, focusing on the algebraic characterization and asymptotic behaviour of stretch factors or equivalently pseudo-Anosov dilatations. We first review the algebraic foundations linking Perron--Frobenius theory, multi-variable Euclidean algorithms, and Lind's characterization of topological entropy for shifts of finite type, emphasizing recent constraints on the Perron--Frobenius degree of expansion constants. We then relate these dynamical invariants to the combinatorial world, examining train tracks for surfaces and automorphisms of free groups. Finally, we briefly review the relevance of fibered 3-manifolds, detailing how the Thurston norm, Fried's concavity theorem over fibered cones, and McMullen's Teichmüller polynomial come into play.

Explore related subjects

math.GT math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Ahmad Rafiqi, Ayberk Zeytin. 2026-10-09. Thurston's Legacy at the Interface of Surface Dynamics, Train Tracks, and Arithmetic. https://arxiv.org/abs/2610.13347

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Counterexample to Three Conjectures of Luo

We give a single counterexample to Conjectures 1--3 as stated in Luo's paper: a minimal two-tetrahedron triangulation of the lens space L(3,1) with a null-homotopic loop edge. The proof uses quadrilateral matching equations, an explicit calculation of shape parameters, and known results on volume maximization.

math.GT↗

arXiv2026

Properness and finiteness of totally geodesic submanifolds in the convex core

We study totally geodesic submanifolds in the convex core of geometrically finite rank-one locally symmetric manifolds. Although the infinite-volume setting can exhibit highly complicated behavior, including geodesic planes with fractal closures, we show that a strong rigidity persists inside the convex core. This rigidity has striking consequences in the infinite volume setting: every maximal totally geodesic submanifold of dimension at least two contained in the convex core is properly immersed and has finite volume, and only finitely many such submanifolds can occur. These results stand in sharp contrast to the behavior in the finite-volume setting. Moreover, combining this finiteness result with the work of Bader-Fisher-Miller-Stover and of Gromov-Schoen, we deduce that any geometrically finite rank-one manifold with infinitely many maximal totally geodesic submanifolds of dimension at least two and of finite volume must be arithmetic.

math.GT↗

arXiv2026

On the Kashaev--Luo--Vartanov Partition Function for Cusped 3-Manifolds: Asymptotics and a squared norm property

We study the asymptotic and TQFT-type properties of the Kashaev--Luo--Vartanov (KLV) partition function. We show that the partition function depends on the prescribed angle structure only through its peripheral angular holonomies. For any geometric triangulation of a cusped 3-manifold realizing a possibly incomplete hyperbolic structure, we express the exponential decay rate and the 1-loop term of the partition function in terms of, respectively, the volume and the adjoint twisted Reidemeister torsion of the corresponding incomplete hyperbolic structure. Consequently, these asymptotic formulas hold for any angle structure having the same peripheral angular holonomies as the geometric one. We further introduce a Reshetikhin--Turaev-type function associated with certain admissible Neumann--Zagier data, extending the Jones function in Teichmüller TQFT previously studied by Ben Aribi and the author. We show that, after a combinatorial normalization, for any triangulation, the KLV partition function can be expressed as a weighted integral of the squared norm of the Reshetikhin--Turaev-type function, where the weight is determined by the angular holonomies of peripheral curves. In particular, for FAMED triangulations, the Reshetikhin--Turaev-type function agrees with the Jones function, so that the normalized KLV partition function is obtained by integrating the squared norm of the integrand defining the Teichmüller TQFT partition function. This provides a noncompact analogue of the relationship between Turaev--Viro and Reshetikhin--Turaev invariants.

math.GT↗