arXiv subjects
Publications and source records attributed to Rémi Coulon.
We construct continuum many infinite, simple, characteristic quotients of non-abelian free groups, answering a 1978 question of James Wiegold. The method is very flexible, allowing to impose certain properties on the quotients, to generalize the construction to large classes of groups with hyperbolic features, and to produce quasi-isometrically diverse examples.
We study the growth spectrum of groups acting on hyperbolic spaces, i.e.\ the set of exponential growth rates achieved by subgroups. For a finitely generated free group or a surface group acting convex-cocompactly on a proper geodesic hyperbolic metric space, we prove that the growth spectrum is the full interval $[0, ω_G]$. For any hyperbolic group, we prove that the growth spectrum contains a large interval $[0, ω_{\mathcal{F}}]$ where $ω_{\mathcal{F}} \geq ω_G / 2$, with strict inequality when the action is divergent. In the case of the Cayley graph of a free group, we also present an approach via the non-backtracking matrix of the configuration model, connecting the density of growth rates to a spectral concentration result for random graphs.
Let $G$ be a toral relatively hyperbolic group, and let $φ\in\mathrm{Aut}(G)$. We prove that, under iteration of $φ$, the conjugacy length $||φ^n(g)||$ of every element $g\in G$ grows like $n^dλ^n$ for some $d\in\mathbb{N}$ and some algebraic integer $λ\geq 1$. For a given $φ$, only finitely many values of $d$ and $λ$ occur as $g$ varies in $G$. The same statements hold for the growth of the word length $|φ^n(g)|$. For $G$ hyperbolic, we generalize polynomial subgroups: we show that, for a given growth type $n^dλ^n$ other than $1$, there is a malnormal family of quasiconvex subgroups $K_1,\dots,K_p$ such that a conjugacy class $[g]$ grows at most like $n^dλ^n$ if and only if $g$ is conjugate into one of the subgroups $K_i$.
We develop methods to control the first-order theory of groups arising as certain direct limits of torsion-free hyperbolic groups, answering several questions in the literature. We construct simple torsion-free Tarski monsters $Γ$ (non-abelian groups whose non-trivial, proper subgroups are infinite cyclic) that are $\exists \forall \exists$-elementarily embedded into $Γ\ast \mathbf{Z}$. In particular, such $Γ$ have the same two-quantifier theory as $Γ\ast \mathbf{Z}$, and hence the same positive theory as a non-abelian free group. All previously known examples of groups with the same positive theory as the free group admit a non-elementary action on a hyperbolic space, while our examples cannot act on a hyperbolic space with a loxodromic element. Along the way, we solve the one-quantifier Knight conjecture for random quotients of arbitrary torsion-free, non-elementary, hyperbolic groups in the few-relator model.
In this article we investigate the dynamical properties of the geodesic flow for a proper metric space endowed with a proper action by isometries of a group with a contracting element. We show that the existence of a contracting isometry is a sufficient evidence of negative curvature to carry in this context various results borrowed from hyperbolic geometry. In particular, we extend the so-called Hopf-Tsuji-Sullivan dichotomy proving that the geodesic flow is either conservative and ergodic or dissipative.
Let $Γ$ be the fundamental group of a closed, orientable, hyperbolic surface $S$. The $n$-power quotient, $Γ(n)$, is the quotient of $Γ$ by the $n$th powers of simple closed curves. We prove an analogue of the Dehn--Nielsen--Baer theorem for suitable large values of $n$: the outer automorphism group of $Γ(n)$ is isomorphic to the quotient of the extended mapping class group of $S$ by $n$th powers of Dehn twists. There is also a corresponding description of the automorphism group as the quotient of the extended mapping class group of the corresponding once-punctured surface, and we relate these groups via a Birman-type exact sequence. Along the way, and as consequences, we prove structural properties of $Γ(n)$ for suitable large values of $n$, including: $Γ(n)$ is virtually torsion-free, acylindrically hyperbolic, infinitely presented, with solvable word problem and finite asymptotic dimension.
This article is a first step in the study of equations in periodic groups. As an application, we study the structure of periodic quotients of hyperbolic groups. We investigate for instance the Hopf and co-Hopf properties, the isomorphism problem, the existence of free splittings, etc. We also consider the automorphism group of such periodic groups.
Using Patterson-Sullivan measures we investigate growth problems for groups acting on a metric space with a strongly contracting element.
Mathematical understanding is built in many ways. Among these, illustration has been a companion and tool for research for as long as research has taken place. We use the term illustration to encompass any way one might bring a mathematical idea into physical form or experience, including hand-made diagrams or models, computer visualization, 3D printing, and virtual reality, among many others. The very process of illustration itself challenges our mathematical understanding and forces us to answer questions we may not have posed otherwise. It can even make mathematics an experimental science, in which immersive exploration of data and representations drive the cycle of problem, conjecture, and proof. Today, modern technology for the first time places the production of highly complicated models within the reach of many individual mathematicians. Here, we sketch the rich history of illustration, highlight important recent examples of its contribution to research, and examine how it can be viewed as a discipline in its own right.
Given a periodic quotient of a torsion-free hyperbolic group, we provide a fine lower estimate of the growth function of any sub-semi-group. This generalizes results of Razborov and Safin for free groups.
We describe algorithms that produce accurate real-time interactive in-space views of the eight Thurston geometries using ray-marching. We give a theoretical framework for our algorithms, independent of the geometry involved. In addition to scenes within a geometry $X$, we also consider scenes within quotient manifolds and orbifolds $X / Γ$. We adapt the Phong lighting model to non-euclidean geometries. The most difficult part of this is the calculation of light intensity, which relates to the area density of geodesic spheres. We also give extensive practical details for each geometry.
We give a new proof that free Burnside groups of sufficiently large even exponents are infinite. The method is very flexible and can also be used to study (partially) periodic quotients of any group which admits an action on a hyperbolic space satisfying a weak form of acylindricity.
This article presents virtual reality software designed to explore the Sol geometry. The simulation is available on 3-dimensional.space/sol.html
We describe a method of rendering real-time scenes in Nil geometry, and use this to give an expository account of some interesting geometric phenomena. You can play around with the simulation at www.3-dimensional.space/nil.html.
In this note we produce examples of outer automorphisms of finitely generated groups which have exotic behaviors in terms of growth of conjugacy classes.
We develop a version of small cancellation theory in the variety of Burnside groups. More precisely, we show that there exists a critical exponent $n_0$ such that for every odd integer $n\geq n_0$, the well-known classical $C'(1/6)$-small cancellation theory, as well as its graphical generalization and its version for free products, produce examples of infinite $n$-periodic groups. Our result gives a powerful tool for producing (uncountable collections of) examples of $n$-periodic groups with prescribed properties. It can be applied without any prior knowledge in the subject of $n$-periodic groups. As applications, we show the undecidability of Markov properties in classes of $n$-periodic groups, we produce $n$-periodic groups whose Cayley graph contains an embedded expander graphs, and we give an $n$-periodic version of the Rips construction. We also obtain simpler proofs of some known results like the existence of uncountably many finitely generated $n$-periodic groups and the SQ-universality (in the class of $n$-periodic groups) of free Burnside groups.
Let $Γ'<Γ$ be two discrete groups acting properly by isometries on a Gromov-hyperbolic space $X$. We prove that their critical exponents coincide if and only if $Γ'$ is co-amenable in $Γ$, under the assumption that the action of $Γ$ on $X$ is strongly positively recurrent, i.e. has a growth gap at infinity. This generalizes all previously known results on this question, which required either $X$ to be the real hyperbolic space and $Γ$ geometrically finite, or $X$ Gromov hyperbolic and $Γ$ cocompact. This result is optimal: we provide several counterexamples when the action is not strongly positively recurrent.
We prove a general version of the amenability conjecture in the unified setting of a Gromov hyperbolic group G acting properly cocompactly either on its Cayley graph, or on a CAT(-1)-space. Namely, for any subgroup H of G, we show that H is co-amenable in G if and only if their exponential growth rates (with respect to the prescribed action) coincide. For this, we prove a quantified, representation-theoretical version of Stadlbauer's amenability criterion for group extensions of a topologically transitive subshift of finite type, in terms of the spectral radii of the classical Ruelle transfer operator and its corresponding extension. As a consequence, we are able to show that, in our enlarged context, there is a gap between the exponential growth rate of a group with Kazhdan's property (T) and the ones of its infinite index subgroups. This also generalizes a well-known theorem of Corlette for lattices of the quaternionic hyperbolic space or the Cayley hyperbolic plane.