Searcharxiv⌕ Search

arXiv subjects

Ser Peow Tan

Publications and source records attributed to Ser Peow Tan.

At least 19 recordsLinked to original sources
arXiv2026

Equidistribution of partial coverings from closed geodesics

Given a finite volume hyperbolic surface, a fundamental polygon and an oriented closed geodesic, we associate a partial covering of the surface. We prove that given a sequence of collections of oriented closed geodesics equidistributing in the unit tangent bundle then the associated partial coverings also equidistribute. In the case of the modular group, this yields an alternative proof of an equidistribution result due to Duke, Imamoglu, and Tóth.

math.GT↗
arXiv2026

Optimal Farey sequence for the Congruence subgroup $Γ_0(2^{n})$

We prove that $Γ_0(2^n)$ ($n\ge2$) has a Farey sequence $\{e_i\}$ such that $e_i \le 2^{n-1}$ for all $e_i$. The above upper bound is optimal, and there exists a unique $j$ such that $e_j= 2^{n-1} $. For each $e_i$, there exists a unique $a_i$ such that $\{ a_i/e_i\}\cup \{\infty\}$ is the set of ideal vertices of a fundamental domain of $Γ_0(2^n)$ whose side-pairings give a set of independent generators of $Γ_0(2^n)$.

math.NT↗
arXiv2023

Optimal independent generating system for the congruence subgroups $Γ_0(p)$ and $Γ_0(p^2)$

Let $n$ be a prime or its square. We prove that the congruence subgroup $Γ_0(n)$ admits a free product decomposition into cyclic factors in such a way that the $(2,1)$-component of each cyclic generator is either $n$ or $0$, answering a conjecture of Kulkarni. We can also require that the Frobenius norm of each generator is less than $2n-1$. A crucial observation is that if $P$ denotes the convex hull of the extended Farey sequence of order $\lfloor \sqrt{n} \rfloor$ in the hyperbolic plane $\mathbb{H}^2$, then the projection $π: \mathbb{H}^2\to \mathbb{H}^2/Γ_0(n)$ is injective on the interior of $P$ and each connected component of $π(\mathbb{H}^2)\setminusπ(P)$ is either an order-three cone of area $π/3$ or an ideal triangle. Denoting by $m(Γ_0(n))$ the minimum of the largest denominator in the cusp set of $Q$ where $Q$ ranges over all possible special (fundamental) polygons for $Γ_0(n)$, we establish the inequality $ \lfloor \sqrt{n} \rfloor \le m(Γ_0(n))\le \lfloor \sqrt{4n/3} \rfloor$, and completely characterize the cases in which the bounds are achieved. We also prove analogous results when $n$ is the multiplication of two sufficiently close odd primes.

math.NT↗
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↗
arXiv2021

Shapes of hyperbolic triangles and once-punctured torus groups

Let $Δ$ be a hyperbolic triangle with a fixed area $φ$. We prove that for all but countably many $φ$, generic choices of $Δ$ have the property that the group generated by the $π$--rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all $φ\in(0,π)\setminus\mathbb{Q}π$, a dense set of triangles does afford nontrivial relations, which in the generic case map to hyperbolic translations. To establish this fact, we study the deformation space $\mathfrak{C}_θ$ of singular hyperbolic metrics on a torus with a single cone point of angle $θ=2(π-φ)$, and answer an analogous question for the holonomy map $ρ_ξ$ of such a hyperbolic structure $ξ$. In an appendix by X.~Gao, concrete examples of $θ$ and $ξ\in\mathfrak{C}_θ$ are given where the image of each $ρ_ξ$ is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3--manifolds.

math.GT↗
arXiv2020

Hyperbolic jigsaws and families of pseudomodular groups II

In our previous paper, we introduced a hyperbolic jigsaw construction and constructed infinitely many non-commensurable, non-uniform, non-arithmetic lattices of $\mathrm{PSL}(2, \mathbb{R})$ with cusp set $\mathbb{Q} \cup \{\infty\}$ (called pseudomodular groups by Long and Reid), thus answering a question posed by Long and Reid. In this paper, we continue with our study of these jigsaw groups exploring questions of arithmeticity, pseudomodularity, and also related pseudo-euclidean and continued fraction algorithms arising from these groups. We also answer another question of Long and Reid by demonstrating a recursive formula for the tessellation of the hyperbolic plane arising from Weierstrass groups which generalizes the well-known "Farey addition" used to generate the Farey tessellation.

math.GT↗
arXiv2020

Dilogarithm identities after Bridgeman

Following Bridgeman, we demonstrate several families of infinite dilogarithm identities associated with Fibonacci numbers, Lucas numbers, convergents of continued fractions of even periods, and terms arising from various recurrence relations.

math.GT↗
arXiv2020

Prime orthogeodesics, concave cores and families of identities on hyperbolic surfaces

We prove and explore a family of identities relating lengths of curves and orthogeodesics of hyperbolic surfaces. These identities hold over a large space of metrics including ones with hyperbolic cone points, and in particular, show how to extend a result of the first author to surfaces with cusps. One of the main ingredients in the approach is a partition of the set of orthogeodesics into sets depending on their dynamical behavior, which can be understood geometrically by relating them to geodesics on orbifold surfaces. These orbifold surfaces turn out to be exactly on the boundary of the space in which the underlying identity holds.

math.GT↗
arXiv2020

Measuring pants

We investigate the terms arising in an identity for hyperbolic surfaces proved by Luo and Tan, namely showing that they vary monotonically in terms of lengths and that they verify certain convexity properties. Using these properties, we deduce two results. As a first application, we show how to deduce a theorem of Thurston which states, in particular for closed hyperbolic surfaces, that if a simple length spectrum "dominates" another, then in fact the two surfaces are isometric. As a second application, we show how to find upper bounds on the number of pairs of pants of bounded length that only depend on the boundary length and the topology of the surface.

math.GT↗
arXiv2018

Carrier graphs for representations of the rank two free group into isometries of hyperbolic three space

Carrier graphs were first introduced for closed hyperbolic 3-manifolds by White. In this paper, we first generalize this definition to carrier graphs for representations of a rank two free group into the isometry group of hyperbolic three space. Then we prove the existence and the finiteness of minimal carrier graphs for those representations which are discrete, faithful and geometrically finite, and more generally, those that satisfy certain finiteness conditions first introduced by Bowditch.

math.GT↗
arXiv2017

Hyperbolic jigsaws and families of pseudomodular groups I

We show that there are infinitely many commensurability classes of pseudomodular groups, thus answering a question raised by Long and Reid. These are Fuchsian groups whose cusp set is all of the rationals but which are not commensurable to the modular group. We do this by introducing a general construction for the fundamental domains of Fuchsian groups obtained by gluing together marked ideal triangular tiles, which we call hyperbolic jigsaw groups.

math.GT↗
arXiv2016

Automorphisms of two-generator free groups and spaces of isometric actions on the hyperbolic plane

The automorphisms of a two-generator free group acting on the space of orientation-preserving isometric actions of on hyperbolic 3-space defines a dynamical system. Those actions which preserve a hyperbolic plane but not an orientation on that plane is an invariant subsystem, which reduces to an action on R^3 by polynomial automorphisms preserving the cubic polynomial and an area form on the level surfaces. We describe the dynamical decomposition of this action. The domain of discontinuity of this action corresponds to geometric structures: either complete hyperbolic structures on the 2-holed cross-surface (projective plane) with cusps and funnels, or complete hyperbolic structures on a one-holed Klein bottle, or hyperbolic structures on a Klein bottle with one conical singularity. The action is ergodic on the complement of the orbit of the Fricke space of the 2-holed cross-surface, and we show that the orbit of the generalized Fricke space of the one-holed Klein bottle is open and dense.

math.DS↗
arXiv2015

Polynomial automorphisms of C^n preserving the Markoff-Hurwitz polynomial

We study the action of the group of polynomial automorphisms of C^n (n>2) which preserve the Markoff-Hurwitz polynomial H(x):= x_1^2 + x_2^2 + ... + x_n^2 - x_1 x_2 ... x_n. Our main results include the determination of the group, the description of a non-empty open subset of C^n on which the group acts properly discontinuously (domain of discontinuity), and identities for the orbit of points in the domain of discontinuity.

math.GT↗
arXiv2014

The diagonal slice of Schottky space

An irreducible representation of the free group on two generators X,Y into SL(2,C) is determined up to conjugation by the traces of X,Y and XY. We study the diagonal slice of representations for which X,Y and XY have equal trace. Using the three-fold symmetry and Keen-Series pleating rays we locate those groups which are free and discrete, in which case the resulting hyperbolic manifold is a genus-2 handlebody. We also compute the Bowditch set, consisting of those representations for which no primitive elements in the group generated by X,Y are parabolic or elliptic, and at most finitely many have trace with absolute value at most 2. In contrast to the quasifuchsian punctured torus groups originally studied by Bowditch, computer graphics show that this set is significantly different from the discreteness locus.

math.GT↗
arXiv2014

A new identity for SL(2,C)-characters of the once punctured torus group

We obtain new variations of the original McShane identity for those SL(2,C)-representations of the once punctured torus group which satisfy the Bowditch conditions, and also for those fixed up to conjugacy by an Anosov mapping class of the torus and satisfying the relative Bowditch conditions.

math.GT↗
arXiv2013

Identities on Hyperbolic Manifolds

In this survey, we discuss four classes of identities due principally to Basmajian, McShane, Bridgeman-Kahn and Luo-Tan on hyperbolic manifolds and provide a unified approach for proving them. We also elucidate on the connections between the various identities.

math.GT↗
arXiv2013

On the character variety of the four-holed sphere

We study the (relative) SL(2,C) character varieties of the four-holed sphere and the action of the mapping class group on it. We describe a domain of discontinuity for this action, and, in the case of real characters, show that this domain of discontinuity may be non-empty on the components where the relative euler class is non-maximal.

math.GT↗