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Christopher-Lloyd Simon

Publications and source records attributed to Christopher-Lloyd Simon.

14 recordsLinked to original sources
arXiv2026

Transcendence of simple geodesics on finite modular covers

The real projective line $\mathbb{R}\mathbf{P}^1$ is the boundary of $\mathbf{HP}=\{z\in \mathbb{C}\colon \Im(z)>0\}$, a model of the hyperbolic plane whose space of geodesics identifies with $\mathcal{G}(\mathbf{HP})=\mathbb{R}\mathbf{P}^1 \times \mathbb{R}\mathbf{P}^1 \setminus \mathrm{diagonal}$. The modular group $\Gamma=\operatorname{PSL}_2(\mathbb{Z})$ acts on $\mathbf{HP}$ with quotient the modular orbifold $\mathbf{M}=\Gamma\backslash \mathbf{HP}$. Consider a finite-index subgroup of the modular group $\Gamma^\prime \subset \Gamma = \operatorname{PSL}_2(\mathbb{Z})$ corresponding to a finite cover $\mathbf{M} \to \mathbf{M}^\prime$. A geodesic $(\xi^-,\xi^+)\in \mathcal{G}(\mathbf{HP})$ projects $\bmod{\Gamma^\prime}$ to a geodesic $\xi^\prime \subset \mathbf{M}^\prime$. We show that if $\xi^\prime$ is simple, then $\xi^+$ is either rational or quadratic or transcendental. In the transcendental case, we obtain bounds on the Mahler measures and show that those can be improved for geodesics fixed by pseud-Anosov maps. Finally, we also explain in detail why all this was known for the modular torus cover associated to the derived subgroup $\Gamma^\prime = [\Gamma, \Gamma]$.

math.NT↗
arXiv2026

The variety of group actions on all algebraic real hyperbolic spaces

For a cardinal $\kappa$, denote by $\mathbf{H}^\kappa$ the algebraic real hyperbolic space of dimension $\kappa$. For a topological group $\Gamma$, we study the set of continuous representations $\Gamma \to \operatorname{Isom}(\mathbf{H}^\kappa)$ up to continuous self-representations $\operatorname{Isom}(\mathbf{H}^\kappa)\to \operatorname{Isom}(\mathbf{H}^\kappa)$. The novelty of this work relies in considering simultaneously all cardinals, finite or infinite. We will endow this set of classes of representations with a natural topology, and show that this character variety is compact. This will also enable us to recover all previous compactifications of actions on $\mathbf{H}^n$ by certain actions on real trees for the equivariant Gromov-Hausdorff topology. A class of representations recovers in particular the homothety class of its marked length spectrum. We will define the notion of algebraic cross-ratio and prove a GNS-embedding result, enabling us to generalize some rigidity properties of the marked length spectrum. We will also introduce a notion of abstract cross-ratio, and use it to show that a wide class of groups $\Gamma$ (characterized by the existence of what we call a $3$-full action on a $\operatorname{CAT}(-1)$-space) admit at most one class of irreducible representations into $\operatorname{Isom}(\mathbf{H}^\kappa)$ whose boundedness properties are controlled by those of $(X,d)$. We will apply this to topological groups $\Gamma$ such as the isometry group $\operatorname{Isom}(\mathbf{H}^\kappa)$ itself, the automorphism group $\operatorname{Aut}(T_\omega)$ of the simplicial tree with countably infinite valency, and the automorphism group $\operatorname{PGL}_2(\mathbb{K}, \lvert\cdot \rvert)$ of the projective line over a non-Archimedean field.

math.MG↗
arXiv2026

The complexity of pinning simple multiloops

A multiloop with $s\in \mathbb{N}$ strands is a generic immersion $\gamma\colon \sqcup_1^s \mathbb{S}^1 \looparrowright \Sigma$ of the union of $s$ circles into a surface $\Sigma$, considered up to homeomorphisms. A pinning set of $\gamma$ is a set of points $P\subset \Sigma\setminus \operatorname{im}(\gamma)$, such that in the punctured surface $\Sigma \setminus P$, the immersion $\gamma$ has the minimal number of double points in its homotopy class. Its pinning number $\varpi(\gamma)$ is the minimum cardinal of its pinning sets. In any fixed orientable surface $\Sigma$, the pinning problem which given a multiloop $\gamma$ and $k\in \mathbb{N}$ decides whether $\varpi(\gamma)\le k$ has been show to be NP-complete, even in restrictions to loops (with $s=1$ strand). In this work we study the complexity of the pinning problem in restriction to multiloops whose strands are simple (embedded circles). We show that in any fixed oriented surface $\Sigma$, the problem is in P when $s\leq 3$ and NP-complete when $s\geq 20$, and present some follow-up questions and conjectures.

math.GT↗
arXiv2025

Hyperbolic Simplices of Maximal Inradius

For $n\in \mathbb{N}$, consider a hyperbolic $n$-dimensional simplex $\Delta$, defined by $1+n$ points in the compactified hyperbolic space $\mathbf{H}^n \sqcup \partial \mathbf{H}^n$. For each integer $m\le n$, denote $\delta^n_m(\Delta)\in [0,+\infty]$ the Hausdorff distance between its skeleta of dimensions $n$ and $m$. In particular, $\delta^n_{n-1}(\Delta)$ is its inradius. The maximum of $\delta^n_m(\Delta)$ over $\Delta\in (\mathbf{H}^n \sqcup \partial \mathbf{H}^n)^{1+n}$ is denoted $\mu^n_m\in [0,+\infty]$. We first show that $\Delta$ has maximal inradius $\delta^n_{n-1}(\Delta)=\mu^n_m$ if and only if its is (total) ideal and regular; for which the inradius is given by $\tanh \mu^n_{n-1} = 1/n$. We deduce that $\Delta$ has maximal $\delta^n_{n-1}(\Delta)=\mu^n_m$ if and only if it is (total) ideal and regular. We compute that the maximal distance to the $1$-skeleton $\mu^n_1$ is given by $\left(\tanh \mu^n_1\right)^2 = (n-1)/(2n)$ and deduce that those are uniformly bounded by $\lim_{n} \mu^n_1 = \log(1+\sqrt{2})$.

math.MG↗
arXiv2025

Isogenies of minimal Cantor systems: from Sturmian to Denjoy and interval exchanges

This work is motivated by the study of continued fraction expansions of real numbers: we describe in dynamical terms their orbits under the action of $\mathrm{PGL}_2(\mathbb{Q})$. A real number gives rise to a Sturmian system encoding a rotation of the circle. It is well known that $\mathrm{PGL}_2(\mathbb{Z})$-equivalence of real numbers, characterized by the tails of their continued fraction expansions, amounts to flow equivalence of Sturmian systems. We show that the multiplicative action of $m\in \mathbb{Z}$ on a real number corresponds to taking the $m$th-power followed by what we call an infinitesimal 2-asymptotic factor of its Sturmian system. This leads us to introduce the notion of isogeny between zero-dimensional systems: it combines virtual flow equivalences and infinitesimal asymptotic equivalences. We develop tools for classifying systems up to isogeny involving cohomological invariants and states. We then use this to give a complete description of $\mathrm{PSL}_2(\mathbb{Q})$-equivalence of real numbers in terms of Sturmian systems. We classify Denjoy systems up to isogenies within this class via the action of $\mathrm{PGL}_{2}(\mathbb{Q})$ on their invariants. We also investigate eventual flow equivalence of Sturmian systems: we show that for non-quadratic parameters it amounts to topological conjugacy and for quadratic parameters it implies total flow equivalence and other arithmetic constraints. In another direction, we consider interval exchanges satisfying Keane's condition. We characterize flow equivalence in terms of interval-induced subsystems (or the tails of their paths in the bilateral Rauzy induction diagram). Finally we find rational invariants for isogeny involving the length modules and SAF invariants of the associated ergodic measures. This leads to a conjecture for their classification up to isogeny, which we prove in the totally ergodic case.

math.DS↗
arXiv2024

Computing the degreewidth of a digraph is hard

Given a digraph, an ordering of its vertices defines a backedge graph, namely the undirected graph whose edges correspond to the arcs pointing backwards with respect to the order. The degreewidth of a digraph is the minimum over all ordering of the maximum degree of the backedge graph. We answer an open question by Keeney and Lokshtanov [WG 2024], proving that it is \NP-hard to determine whether an oriented graph has degreewidth at most $1$, which settles the last open case for oriented graphs. We complement this result with a general discussion on parameters defined using backedge graphs and their relations to classical parameters.

math.CO↗
arXiv2024

The pinning ideal of a multiloop

A multiloop $\gamma\colon \sqcup_1^s \mathbb{S}^1 \looparrowright \mathbb{F}$ is a generic immersion of a finite union of circles into an oriented surface, considered up to homeomorphisms. A pinning set is a set of points $P\subset \mathbb{F}\setminus \operatorname{im}(\gamma)$, such that in the punctured surface $\mathbb{F} \setminus P$, the immersion $\gamma$ has the minimal number of double points in its homotopy class. The collection of pinning sets of $\gamma$ forms a poset under inclusion called the pinning ideal $\mathcal{PI}(\gamma)$ which is endowed with the cardinal function whose minimum defines the pinning number $\varpi(\gamma)$. We show that the decision problem associated to computing the pinning number of a multiloop is \textsf{NP}-complete, even for loops in the sphere. We give two proofs that it is \textsf{NP}: First, we implement a polynomial algorithm to check if a point-set is pinning, adapting methods of Birman--Series and Cohen--Lustig for computing intersection numbers of curves in surfaces. Second, for loops in the sphere we reduce the problem in polynomial time to a variant of boolean satisfiability by applying a theorem of Hass--Scott characterizing taut loops, and adapting algorithms of Blank and Shor--Van Wyk which decide when a curve in the plane bounds an immersed disc. To show that it is \textsf{NP}-hard we reduce the vertex cover problem for graphs to the pinning problem for plane loops. We use our algorithms to compute the pinning ideals for $\approx 1000$ of the smallest multiloops in the sphere, available in the online catalog LooPindex.

math.GT↗
arXiv2024

Geometry and Transcendence of the Hexponential

The modular group $\operatorname{PSL}_2(\mathbb{Z})$ acts on the upper-half plane $\mathbb{HP}$ with quotient the modular orbifold, uniformized by the function $\mathfrak{j} \colon \mathbb{HP}\to \mathbb{C}$. We first show that second derived subgroup $\operatorname{PSL}_2(\mathbb{Z})''$ corresponds to a $\mathbb{Z}^2\rtimes \mathbb{Z}/6$ Galois cover of the modular orbifold by a hexpunctured plane, uniformized by the hexponential map $\operatorname{hexp} \colon \mathbb{HP} \to \mathbb{C} \setminus (ω_0\mathbb{Z}[j])$, which is a primitive of $Cη^4$ where $ω_0\in i\mathbb{R}$ and $C\in \mathbb{R}$ are explicit constants and $η$ is Dedekind eta function. We describe the values of the cusp-compactification $\partial \operatorname{hexp}\colon \mathbb{QP}^1\to ω_0 \mathbb{Z}[j]$. After defining the radial-compactification $\operatorname{Shexp} \colon \mathscr{R} \to \mathbb{R}/(2π\mathbb{Z})$, we construct a simple section $\operatorname{InSh} \colon \mathbb{R}/(2π\mathbb{Z}) \to \mathscr{S} \bmod{\operatorname{PSL}_2(\mathbb{Z})'}$ where $\mathscr{S} \subset \mathbb{RP}^1$ is a set of numbers whose continued fraction expansions arise from Sturmian sequences, which contains the set $\mathscr{M}$ of Markov quadratic irrationals as those numbers arising from periodic Sturmian sequences. We will show that the values of $\operatorname{InSh}$ are either Markov quadratic irrationals or transcendental. Finally we provide a continued fraction expansion for $\operatorname{hexp}$, and discuss its monodromy.

math.NT↗
arXiv2023

Loops in surfaces, chord diagrams, interlace graphs: operad factorisations and generating grammars

A filoop is a generic immersion of a circle in a closed oriented surface, whose complement is a disjoint union of discs, considered up to orientation preserving diffeomorphisms. It gives rise to a chord diagram C which has an interlace graph G, called a chordiagraph. For a graph G with even degrees, we compute a quantity mg(G) which yields, for every chord diagram $C$ with interlace graph G, the minimal genus of filoops with chord diagram C. If mg(G)=0 then C admits exactly two framings of genus 0, corresponding to spheriloops. After recalling the Cunningham factorisation of connected graphs, we describe a canonical factorisation of filoops into spheric sums followed by toric sums, for which the genus is additive. This is analogous to the factorisation of compact connected 3-manifolds along spheres and tori. We describe unambiguous context-sensitive grammars generating the set of all graphs and with mg(G)=0 and deduce stability properties with respect to spheric and toric factorisations. Similar results hold for chordiagraphs with mg(G) = 0 and their corresponding spheriloops.

math.GT↗
arXiv2022

Linking numbers of modular knots

The modular group PSL(2;Z) acts on the hyperbolic plane HP with quotient the modular surface M, whose unit tangent bundle U is a 3-manifold homeomorphic to the complement of the trefoil knot in the 3-sphere. The hyperbolic conjugacy classes of PSL(2;Z) correspond to the closed oriented geodesics in M. Those lift to the periodic orbits for the geodesic flow in U, which define the modular knots. The linking numbers between modular knots and the trefoil is well understood. Indeed, Etienne Ghys showed in 2006 that they are given by the Rademacher invariant of the corresponding conjugacy classes. The Rademacher function is a homogeneous quasi-morphism of PSL(2;Z) which he had recognised with Jean Barge in 1992 as half the primitive of the bounded euler class. This shed light on the 1987 work of Michael Atiyah concerning the logarithm of the Dedekind eta function which identified it with no less than that six other important functions appearing in diverse areas of mathematics. We are concerned with the linking numbers between modular knots and derive several formulae with arithmetical, combinatorial, topological and group theoretical flavours. In particular we associate to a pair of modular knots a function defined on the character variety of PSL(2;Z), whose limit at the boundary point recovers their linking number. Moreover, we show that the linking number with a modular knot minus that with its inverse yields a homogeneous quasi-morphism on the modular group, and how to extract a free basis out of these. For this we prove that the linking pairing is non degenerate.

math.GT↗
arXiv2022

Conjugacy classes in PSL(2, K)

We first describe, over a field K of characteristic different from 2, the orbits for the adjoint actions of the Lie groups PGL(2, K) and PSL(2, K) on their Lie algebra sl(2, K). While the former are well known, the latter lead to the resolution of generalised Pell-Fermat equations which characterise the corresponding orbit. The synthetic approach enables to change the base field, and we illustrate this picture over the fields with three and five elements, in relation with the geometry of the tetrahedral and icosahedral groups. While the results may appear familiar, they do not seem to be covered in such generality or detail by the existing literature. We apply this discussion to partition the set of PSL(2, Z)-classes of integral binary quadratic forms into groups of PSL(2, K)-classes. When K = C we obtain the class groups of a given discriminant. Then we provide a complete description of their partition into PSL(2, Q)-classes in terms of Hilbert symbols, and relate this to the partition into genera. The results are classical, but our geometrical approach is of independent interest as it may yield new insights into the geometry of Gauss composition, and unify the picture over function fields. Finally we provide a geometric interpretation in the modular orbifold PSL(2, Z) \ H for when two points or two closed geodesics correspond to PSL(2, K)-equivalent quadratic forms, in terms of hyperbolic distances and angles between those modular cycles. These geometric quantities are related to linking numbers of modular knots. Their distribution properties could be studied using the geometry of the quadratic lattice (sl(2, Z), det) but such investigations are not pursued here.

math.GR↗
arXiv2021

Valuations on the character variety: Newton polytopes and Residual Poisson Bracket

We study the space of measured laminations ML on a closed surface from the valuative point of view. We introduce and study a notion of Newton polytope for an algebraic function on the character variety. We prove for instance that trace functions have unit coefficients at the extremal points of their Newton polytope. Then we provide a definition of tangent space at a valuation and show how the Goldman Poisson bracket on the character variety induces a symplectic structure on this valuative model for ML. Finally we identify this symplectic space with previous constructions due to Thurston and Bonahon.

math.GT↗
arXiv2019

Automorphisms of character varieties

We show that the algebraic automorphism group of the SL(2,C) character variety of a closed orientable surface with negative Euler characteristic is a finite extension of its mapping class group. Along the way, we provide a simple characterization of the valuations on the character algebra coming from measured laminations.

math.GT↗