Showing papers in "Groups, Geometry, and Dynamics in 2017"

Morse Boundaries of Proper Geodesic Metric Spaces

Abstract: We introduce a new type of boundary for proper geodesic spaces, called the Morse boundary, that is constructed with rays that identify the "hyperbolic directions" in that space. This boundary is a quasi-isometry invariant and thus produces a well-defined boundary for any finitely generated group. In the case of a proper $\mathrm{CAT}(0)$ space this boundary is the contracting boundary of Charney and Sultan and in the case of a proper Gromov hyperbolic space this boundary is the Gromov boundary. We prove three results about the Morse boundary of Teichm\"uller space. First, we show that the Morse boundary of the mapping class group of a surface is homeomorphic to the Morse boundary of the Teichm\"uller space of that surface. Second, using a result of Leininger and Schleimer, we show that Morse boundaries of Teichm\"uller space can contain spheres of arbitrarily high dimension. Finally, we show that there is an injective continuous map of the Morse boundary of Teichm\"uller space into the Thurston compactification of Teichm\"uller space by projective measured foliations.

Equicontinuous actions of semisimple groups

Abstract: We study equicontinuous actions of semisimple groups and some generalizations. We prove that any such action is universally closed, and in particular proper. We derive various applications, both old and new, including closedness of continuous homomorphisms, nonexistence of weaker topologies, metric ergodicity of transitive actions and vanishing of matrix coefficients for reflexive (more generally: WAP) representations.

Strongly aperiodic subshifts on surface groups

Abstract: We give strongly aperiodic subshifts of finite type on every hyperbolic surface group; more generally, for each pair of expansive primitive symbolic substitution systems with incommensurate growth rates, we construct strongly aperiodic subshifts of finite type on their orbit graphs.

On the genericity of pseudo-Anosov braids II: conjugations to rigid braids

Abstract: We prove that generic elements of braid groups are pseudo-Anosov, in the following sense: in the Cayley graph of the braid group with n $\ge$ 3 strands, with respect to Garside's generating set, we prove that the proportion of pseudo-Anosov braids in the ball of radius l tends to 1 exponentially quickly as l tends to infinity. Moreover, with a similar notion of genericity, we prove that for generic pairs of elements of the braid group, the conjugacy search problem can be solved in quadratic time. The idea behind both results is that generic braids can be conjugated "easily" into a rigid braid.

Real reflections, commutators and cross-ratios in complex hyperbolic space

Abstract: We provide a concrete criterion to determine whether or not two given elements of PU(2,1) can be written as products of real reflections, with one reflection in common. As an application, we show that the Picard modular groups ${\rm PU}(2,1,\mathcal{O}_d)$ with $d=1,2,3,7,11$ are generated by real reflections up to index 1, 2, 4 or 8.

Almost split Kac-Moody groups over ultrametric fields

Abstract: For a split Kac-Moody group G over an ultrametric field K, S. Gaussent and the author defined an ordered affine hovel on which the group acts; it generalizes the Bruhat-Tits building which corresponds to the case when G is reductive. This construction was generalized by C. Charignon to the almost split case when K is a local field. We explain here these constructions with more details and prove many new properties e.g. that the hovel of an almost split Kac-Moody group is an ordered affine hovel, as defined in a previous article.

Coxeter group in Hilbert geometry

Abstract: A theorem of Tits - Vinberg allows to build an action of a Coxeter group $\Gamma$ on a properly convex open set $\Omega$ of the real projective space, thanks to the data $P$ of a polytope and reflection across its facets. We give sufficient conditions for such action to be of finite covolume, convex-cocompact or geometrically finite. We describe an hypothesis that make those conditions necessary. Under this hypothesis, we describe the Zariski closure of $\Gamma$, find the maximal $\Gamma$-invariant convex, when there is a unique $\Gamma$-invariant convex, when the convex $\Omega$ is strictly convex, when we can find a $\Gamma$-invariant convex $\Omega'$ which is strictly convex.

Full groups of Cuntz–Krieger algebras and Higman–Thompson groups

Abstract: In this paper, we will study presentations of the continuous full group $\Gamma_A$ of a one-sided topological Markov shift $(X_A,\sigma_A)$ for an irreducible matrix $A$ with entries in $\{0,1\}$ as a generalization of Higman-Thompson groups $V_N, 1

Invariant random perfect matchings in Cayley graphs

Abstract: We prove that any non-amenable Cayley graph admits a factor of IID perfect matching. We also show that any connected d-regular vertex transitive graph admits a perfect matching. The two results together imply that every Cayley graph admits an invariant random perfect matching. A key step in the proof is a result on graphings that also applies to finite graphs. The finite version says that for any partial matching of a finite regular graph that is a good expander, one can always find an augmenting path whose length is poly-logarithmic in one over the ratio of unmatched vertices.

Independence Tuples and Deninger's Problem

Abstract: Motivated by our results in "Polish Models and Sofic Entropy," we define modified version of the independence tuples for sofic entropy developed by Kerr and Li. These modified version essentially require that the independence sequences give rise to representations weakly contained in the left regular when projected onto the Koopman representation. Using this, we can generalize our previous results for Deninger's Problem. Namely, we can show that if G is a sofic group, and if f is in M_{n}(Z(G)) and is invertible as an operator on l^{2}(G)^{n}, then the Fuglede-Kadison determinant of f is 1 if and only if f is invertible in M_{n}(Z(G)).

On the genericity of pseudo-Anosov braids I: rigid braids

Abstract: We prove that, in the l-ball of the Cayley graph of the braid group with n ≥ 3 strands, the proportion of rigid pseudo-Anosov braids is bounded below independently of l by a positive value.

Which geodesic flows are left-handed?

Abstract: Left-handed flows are 3-dimensional flows which have a particular topological property, namely that every pair of periodic orbits is negatively linked. This property (introduced by Ghys in 2007) implies the existence of as many Bikrhoff sections as possible, and therefore allows to reduce the flow to a suspension in many different ways. It then becomes natural to look for examples. A construction of Birkhoff (1917) suggests that geodesic flows are good candidates. In this conference we determine on which hyperbolic orbifolds is the geodesic flow left-handed: the answer is that yes if the surface is a sphere with three cone points, and no otherwise. dynamical system - geodesic flow - knot - periodic orbit - global section - linking number - fibered knot

Equidistribution, ergodicity and irreducibility in CAT(-1) spaces

Abstract: We prove an equidistribution theorem a la Bader-Muchnik for operator-valued measures associated with boundary representations in the context of discrete groups of isometries of CAT(-1) spaces thanks to an equidistribution theorem of T. Roblin. This result can be viewed as a generalization of Birkhoff's ergodic theorem for quasi invariant measures. In particular, this approach gives a dynamical proof of the fact that boundary representations are irreducible. Moreover, we prove some equidistribution results for conformal densities using elementary techniques from harmonic analysis.

Fibered commensurability and arithmeticity of random mapping tori

Abstract: We consider a random walk on the mapping class group of a surface of finite type. We assume that the random walk is determined by a probability measure whose support is finite and generates a non-elementary subgroup H. We further assume that H is not consisting only of lifts with respect to any one covering. Then we prove that the probability that such a random walk gives a non-minimal mapping class in its fibered commensurability class decays exponentially. As an application of the minimality, we prove that for the case where a surface has at least one puncture, the probability that a random walk gives mapping classes with arithmetic mapping tori decays exponentially. We also prove that a random walk gives rise to asymmetric mapping tori with exponentially high probability for closed case. Mathematics Subject Classification (2010). 20F65, 60G50, 57M50.

Minimal exponential growth rates of metabelian Baumslag–Solitar groups and lamplighter groups

Abstract: We prove that for any prime $p\geq 3$ the minimal exponential growth rate of the Baumslag-Solitar group $BS(1,p)$ and the lamplighter group $\mathcal{L}_p=(\mathbb{Z}/p\mathbb{Z})\wr \mathbb{Z}$ are equal. We also show that for $p=2$ this claim is not true and the growth rate of $BS(1,2)$ is equal to the positive root of $x^3-x^2-2$, whilst the one of the lamplighter group $\mathcal{L}_2$ is equal to the golden ratio $(1+\sqrt5)/2$. The latter value also serves to show that the lower bound of A.Mann from [Mann, Journal of Algebra 326, no. 1 (2011) 208--217] for the growth rates of non-semidirect HNN extensions is optimal.

Chain conditions, elementary amenable groups, and descriptive set theory

Abstract: We first consider three well-known chain conditions in the space of marked groups: the minimal condition on centralizers, the maximal condition on subgroups, and the maximal condition on normal subgroups. For each condition, we produce a characterization in terms of well-founded descriptive-set-theoretic trees. Using these characterizations, we demonstrate that the sets given by these conditions are co-analytic and not Borel in the space of marked groups. We then adapt our techniques to show elementary amenable marked groups may be characterized by well-founded descriptive-set-theoretic trees, and therefore, elementary amenability is equivalent to a chain condition. Our characterization again implies the set of elementary amenable groups is co-analytic and non-Borel. As corollary, we obtain a new, non-constructive, proof of the existence of finitely generated amenable groups that are not elementary amenable.

The infinite simple group V of Richard J. Thompson : presentations by permutations

Abstract: We show that one can naturally describe elements of R. Thompson's finitely presented infinite simple group $V$, known by Thompson to have a presentation with four generators and fourteen relations, as products of permutations analogous to transpositions. This perspective provides an intuitive explanation towards the simplicity of $V$ and also perhaps indicates a reason as to why it was one of the first discovered infinite finitely presented simple groups: it is (in some basic sense) a relative of the finite alternating groups. We find a natural infinite presentation for $V$ as a group generated by these "transpositions," which presentation bears comparison with Dehornoy's infinite presentation and which enables us to develop two small presentations for $V$: a human-interpretable presentation with three generators and eight relations, and a Tietze-derived presentation with two generators and seven relations.

Endomorphisms, train track maps, and fully irreducible monodromies

Abstract: Any endomorphism of a finitely generated free group naturally descends to an injective endomorphism of its stable quotient. In this paper, we prove a geometric incarnation of this phenomenon: namely, that every expanding irreducible train track map inducing an endomorphism of the fundamental group gives rise to an expanding irreducible train track representative of the injective endomorphism of the stable quotient. As an application, we prove that the property of having fully irreducible monodromy for a splitting of a hyperbolic free-by-cyclic group depends only on the component of the BNS-invariant containing the associated homomorphism to the integers.

Decision problems, complexity, traces, and representations

Abstract: In this article, we study connections between representation theory and efficient solutions to the conjugacy problem on infinitely generated groups. The main focus is on the conjugacy problem in conjugacy separable groups, where we measure efficiency in terms of the size of the quotients required to distinguish a distinct pair of conjugacy classes.

Gradings on Lie algebras with applications to infra-nilmanifolds

Abstract: In this paper, we study positive as well as non-negative and non-trivial gradings on finite dimensional Lie algebras. We give a different proof that the existence of such a grading on a Lie algebra is invariant under taking field extensions, a result very recently obtained by Y. Cornulier. Similarly, we prove that given a grading of one these types and a finite group of automorphisms, there always exist a positive grading which is preserved by this group. From these results we conclude that the existence of an expanding map or a non-trivial selfcover on an infra-nilmanifold depends only on the covering Lie group. Another application is the construction of a nilmanifold admitting an Anosov diffeomorphisms but no non-trivial self-covers and in particular no expanding maps, which is the first known example of this type. Let E ⊆ C be a subfield of the complex numbers and n a finite dimensional Lie algebra over E. A grading of the Lie algebra n is a decomposition of n as a direct sum

Construction of minimal skew products of amenable minimal dynamical systems

Abstract: For an amenable minimal topologically free dynamical system $\alpha$ of a group on a compact metrizable space $Z$ and for a compact metrizable space $Y$ satisfying a mild condition, we construct a minimal skew product extension of $\alpha$ on $Z\times Y$. This generalizes a result of Glasner and Weiss. We also study the pure infiniteness of the crossed products of minimal dynamical systems arising from this result. In particular, we give a generalization of a result of R{\o}rdam and Sierakowski.

Embedding mapping class groups into a finite product of trees

Abstract: We prove the equivalence between a relative bottleneck property and being quasi-isometric to a tree-graded space. As a consequence, we deduce that the quasi-trees of spaces defined axiomatically by Bestvina-Bromberg-Fujiwara are quasi-isometric to tree-graded spaces. Using this we prove that mapping class groups quasi-isometrically embed into a finite product of simplicial trees. In particular, these groups have finite Assouad-Nagata dimension, direct embeddings exhibiting p compression exponent 1 for all p 1 and they quasi-isometrically embed into 1.N/. We deduce similar consequences for relatively hyperbolic groups whose parabolic subgroups satisfy such conditions. In obtaining these resultswe also demonstrate that curve complexes of compact surfaces and coned-o graphs of relatively hyperbolic groups admit quasi-isometric embeddings into finite products of trees.

Infinite unicorn paths and Gromov boundaries

Abstract: We extend the notion of unicorn paths between two arcs introduced by Hensel, Przytycki and Webb to the case where we replace one arc with a geodesic asymptotic to a lamination. Using these paths, we give new proofs of the results of Klarreich and Schleimer identifying the Gromov boundaries of the curve graph and the arc graph, respectively, as spaces of laminations.

Curves intersecting exactly once and their dual cube complexes

Abstract: Let Sg denote the closed orientable surface of genus g. We construct exponentially many mapping class group orbits of collections of 2g + 1 simple closed curves on Sg which pairwise intersect exactly once, extending a result of the first author [Aou] and further answering a question of Malestein-Rivin-Theran [MRT]. To distinguish such collections up to the action of the mapping class group, we analyze their dual cube complexes in the sense of Sageev [Sag1]. In particular, we show that for any even k between bg/2c and g, there exists such collections whose dual cube complexes have dimension k, and we prove a simplifying structural theorem for any cube complex dual to a collection of curves on a surface pairwise intersecting at most once.

Finitely presented groups and the Whitehead nightmare

Abstract: We define a `nice representation' of a finitely presented group G as being a non-degenerate essentially surjective simplicial map f from a `nice' space X into a 3-complex associated to a presentation of G, with a strong control over the singularities of f, and such that X is WGSC (weakly geometrically simply connected), meaning that it admits a filtration by simply connected and compact subcomplexes. In this paper we study such representations for a very large class of groups, namely QSF (quasi-simply filtered) groups, where QSF is a topological tameness condition of groups that is similar, but weaker, than WGSC. In particular, we prove that any QSF group admits a WGSC representation which is locally finite, equivariant and whose double point set is closed.

Asymptotic shapes for ergodic families of metrics on nilpotent groups

Abstract: Let be a finitely generated nilpotent group. We consider thr ee closely related prob- lems: (i) the asymptotic cone for an equivariant ergodic family of inner metrics on , generalizing Pansu's theorem; (ii) the limit shapes for First Passage Percolation for general (not necessarily independent) ergodic process on edges of a Cayley graph of ; (iii) a sub-additive ergodic theorem over a general ergodic -action. The limiting objects are gi ven in terms of a Carnot-Caratheodory metric on the graded nilpotent group associated to the Mal'cev completion of . 1. Introduction and statement of the main results Let be a finitely generated nilpotent group. The topic of thi s paper may be viewed from three slightly different perspectives: (i) As a generalization of the result of Pansu (12) showing that the asymptotic cone of an inner right-invariant metric d on is the graded nilpotent Lie group G∞ (associated with the Mal'cev completion G of ) equipped with a certain Carnot-Caratheodory metric d∞. Here we show that if one replaces a single invariant metric d by an equivariant ergodic family {dx | x ∈ X} of inner metrics on , then a.e. ( ,dx,e) has the same asymptotic cone which is the graded nilpotent Lie group G∞ equipped with a fixed Carnot-Caratheodory metric associated to certain averages of the family {dx | x ∈ X}. (ii) As a result about asymptotic shape for First Passage Percolation model over driven by a general ergodic process y (X,m). (The case of independent times was recently studied by Benjamini and Tessera (2)). (iii) As a Subadditive Ergodic Theorem over a general ergodic probability measure preserving (hereafter p.m.p.) action y (X,m). Given a measurable function c : × X → R, satisfying

Minimal models for actions of amenable groups

Abstract: We prove that on a metrizable, compact, zero-dimen- sional space every free action of an amenable group is measurably isomorphic to a minimal G-action with the same, i.e. affinely home- omorphic, simplex of measures.

Almost algebraic actions of algebraic groups and applications to algebraic representations

Abstract: Let G be an algebraic group over a complete separable valued field k. We discuss the dynamics of the G-action on spaces of probability measures on algebraic G-varieties. We show that the stabilizers of measures are almost algebraic and the orbits are separated by open invariant sets. We discuss various applications, including existence results for algebraic representations of amenable ergodic actions. The latter provides an essential technical step in the recent generalization of Margulis-Zimmer super-rigidity phenomenon [BF13].

Isolated orderings on amalgamated free products

Abstract: We show that an amalgamated free product $G*_{A}H$ admits a discrete isolated ordering, under some assumptions of $G,H$ and $A$. This generalizes the author's previous construction of isolated orderings, and unlike known constructions of isolated orderings, can produce an isolated ordering with many non-trivial proper convex subgroups.