Rémi Coulon
In this note we produce examples of outer automorphisms of finitely generated groups which have exotic behaviors in terms of growth of conjugacy classes.
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Growth and displacement of free product automorphisms
本文证明不可约自由群自同构的增长率等于其在Culler-Vogtmann空间上的最小Lipschitz位移，并将此结果推广到可约情形及自由积的相对增长率情形。
We prove that an outer automorphism of the free group is exponentially growing if and only if it induces an outer automorphism of infinite order of free Burnside groups with sufficiently large odd exponent.
We generalize a class of groups defined by Rostislav Grigorchuk to a much larger class of groups, and provide upper and lower bounds for their word growth (they are all of intermediate growth) and period growth (under a small additional condition, they are periodic).
We prove that if F is a finitely generated free group and f:F -> F is an automorphism with polynomial growth of degree d, then there exists a characteristic subgroup S < F of finite index such that the induced automorphism of the abelianisation of S also grows polynomially of degree d. The proof is geometric in nature and makes use of improved relative train track representatives.
We look at the automorphisms of Thompson type groups of piecewise linear homeomorphisms of the real line or circle that use slopes that are integral powers of a fixed integer n with n>2. We show that large numbers of "exotic" automorphisms appear---automorphisms that are represented as conjugation by non-PL homeomorphisms of the real line or circle. This is in contrast to the n=2 case where no such automorphisms appear.
Let $Δ$ be a locally finite thick building of type $\tilde{A}_2$. We show that, if the type-preserving automorphism group $\mathrm{Aut}(Δ)^+$ of $Δ$ is transitive on panels of each type, then either $Δ$ is Bruhat--Tits or $\mathrm{Aut}(Δ)$ is discrete. For $\tilde{A}_2$-buildings which are not panel-transitive but only vertex-transitive, we give additional conditions under which the same conclusion holds. We also find a local condition under which an $\tilde{A}_2$-building is ensured to be exotic (i.e.\ not Bruhat--Tits). It can be used to show that the number of exotic $\tilde{A}_2$-buildings…
We study the minimally displaced set of irreducible automorphisms of a free group. Our main result is the co-compactness of the minimally displaced set of an irreducible automorphism with exponential growth $ϕ$, under the action of the centraliser $C(ϕ)$. As a corollary, we get that the same holds for the action of $<ϕ>$ on $Min(ϕ)$. Finally, we prove that the minimally displaced set of an irreducible automorphism of growth rate one is consisted of a single point.
Centralisers of linear growth automorphisms of free groups
本文研究自由群 Out(F_n) 中线性增长元素的中心化子，通过有限指数子群到等变 McCool 群直积的映射结构，证明其具有 VF 性质并因而有限展示。
Automorphism growth and group decompositions
本文研究通过群分解推断自同构和外自同构在整体群上的增长速率问题。
Given an automorphism of a free group $F_n$, we consider the following invariants: $e$ is the number of exponential strata (an upper bound for the number of different exponential growth rates of conjugacy classes); $d$ is the maximal degree of polynomial growth of conjugacy classes; $R$ is the rank of the fixed subgroup. We determine precisely which triples $(e,d,R)$ may be realized by an automorphism of $F_n$. In particular, the inequality $e\le (3n-2)/4}$ (due to Levitt-Lustig) always holds. In an appendix, we show that any conjugacy class grows like a polynomial times an exponential under i…
For a fully irreducible automorphism ϕof the free group F_k we compute the asymptotics of the intersection number n \mapsto i(T,T'ϕ^n) for trees T,T' in Outer space. We also obtain qualitative information about the geometry of the Guirardel core for the trees T and T'ϕ^n for n large.
Growth of automorphisms of virtually special groups
研究虚拟特殊群的外自同构增长速度，证明其增长为多项式或指数，并构建了类似曲面同胚的分解。
PolExp growth for automorphisms of toral relatively hyperbolic groups
研究toral相对双曲群的自同构的PolExp增长，证明共轭长度和词长的增长形式，并推广多项式子群的性质。
We introduce the notion of `core growth rate' for strongly synchronizing transducers. We explore some elementary properties of the core growth rate and give examples of transducers with exponential core growth rate. We conjecture that all strongly synchronizing transducers which generate an automaton group of infinite order have exponential core growth rate. There is a connection to the group of automorphisms of the one-sided shift. More specifically, the results of this article are related to the question of whether or not there can exist infinite order automorphisms of the one-sided shift wi…
We study lattices on ~A_2 buildings that preserve types, act regularly on each type of edge, and whose vertex stabilizers are cyclic. We show that several of their properties, such as their automorphism group and isomorphism class, can be determined from purely combinatorial data. As a consequence we can show that the number of such lattices (up to isomorphism) grows super-exponentially with the thickness parameter q. We look in more detail at the 3295 lattices with q in {2,3,4,5}. We show that with one exception for each q these are all exotic. For the exotic examples we prove that the automo…
A criterion for quadratic or higher growth of group automorphisms is established which are represented by graph-of-groups automorphisms with certain well specified properties. As a consequence, it is derived (using results of a previous paper of the author) that every partial Dehn twist automorphism of $\FN$ relative to local Dehn twist automorphisms is either an honest Dehn twist automorphism, or else has quadratic growth.
Degree growth for tame automorphisms of an affine quadric threefold
本文利用赋值估计和方复形上的群作用，证明了保持仿射二次三维簇的驯顺自同构的动力次数避开 1 和 4/3 之间的值，并刻画了有限多个此类自同构随机乘积的次数增长指数为正的条件。
Cubulating mapping tori of some polynomial growth free group automorphisms
本文给出了多项式增长自由群自同构映射环自由作用于 CAT(0) 立方复形的充分条件，利用循环层级与立方小消去理论证明随机幂单多项式增长自同构的映射环可被立方化。
We prove that the mapping torus group $\FN \rtimes_α \Z$ of any automorphism $α$ of a free group $\FN$ of finite rank $n \geq 2$ is weakly hyperbolic relative to the canonical (up to conjugation) family $\mathcal H(α)$ of subgroups of $\FN$ which consists of (and contains representatives of all) conjugacy classes that grow polynomially under iteration of $α$. Furthermore, we show that $\FN \rtimes_α \Z$ is strongly hyperbolic relative to the mapping torus of the family $\mathcal H(α)$. As an application, we use a result of Drutu-Sapir to deduce that $\FN \rtimes_α \Z$ has Rapic Decay.
Study of the dynamics of automorphisms of a group is usually focused on their growth and/or finite orbits, including fixed points. In this paper, we introduce properties of a different kind; using somewhat informal language, we call them metric properties. Two principal characteristics of this kind are called here the "curl" and the "flux"; there seems to be very little correlation between these and the growth of an automorphism, which means they are likely to be an essentially new tool for studying automorphisms. We also observe that our definitions of the curl and flux are sufficiently gener…
We show that the horoboundary of outer space for the Lipschitz metric is a quotient of Culler and Morgan's classical boundary, two trees being identified whenever their translation length functions are homothetic in restriction to the set of primitive elements of $F_N$. We identify the set of Busemann points with the set of trees with dense orbits. We also investigate a few properties of the horoboundary of outer space for the backward Lipschitz metric, and show in particular that it is infinite-dimensional when $N\ge 3$. We then use our description of the horoboundary of outer space to derive…
An acylindrically hyperbolic group of non-uniform exponential growth
构造了一个具有非一致指数增长和Kazhdan性质(T)的圆柱双曲群，该群是递增有限秩自由群的自同构群的公共商。
We describe, through the use of Rubin's theorem, the automorphism groups of the Higman-Thompson groups $G_{n,r}$ as groups of specific homeomorphisms of Cantor spaces $\mathfrak{C}_{n,r}$. This continues a thread of research begun by Brin, and extended later by Brin and Guzmán: to characterise the automorphism groups of the `Chameleon groups of Richard Thompson,' as Brin referred to them in 1996. The work here completes the first stage of that twenty-year-old program, containing (amongst other things) a characterisation of the automorphism group of $V$, which was the `last chameleon.' The home…
We obtain sufficient and necessary conditions for the Choquet-Deny theorem to hold in the class of compactly generated totally disconnected locally compact groups of polynomial growth, and in a larger class of totally disconnected generalized $\ov{FC}$-groups. The following conditions turn out to be equivalent when $G$ is a metrizable compactly generated totally disconnected locally compact group of polynomial growth: (i) the Choquet-Deny theorem holds for $G$; (ii) the group of inner automorphisms of $G$ acts distally on $G$; (iii) every inner automorphism of $G$ is distal; (iv) the contracti…
Homology growth of polynomially growing mapping tori
本文研究多项式增长自同构的映射圆柱的同调增长性质，证明其具有廉价重建性质并显示torsion同调增长在Farber序列下消失。
If a finitely generated torsion free group K has the property that all finitely generated subgroups S of K are either small or have growth constant bounded uniformly away from 1 then a non proper HNN extension G of K, that is a semidirect product of K by the integers, has the same property. Here small means cyclic or, if the automorphism has no periodic conjugacy classes, free abelian of bounded rank.
We introduce a quantitative characterization of subgroup alternatives modeled on the Tits alternative in terms of group laws and investigate when this property is preserved under extensions. We develop a framework that lets us expand the classes of groups known to have locally uniform exponential growth to include extensions of either word hyperbolic or right-angled Artin groups by groups with locally uniform exponential growth. From this, we deduce that the automorphism group of a torsion-free one-ended hyperbolic group has locally uniform exponential growth. Our methods also demonstrate that…
The Conjugacy Problem for $Out(F_3)$
本文利用可计算不变量、train track 与映射环面，证明 $Out(F_3)$ 中两类剩余情形的共轭问题可判定，从而解决 $Out(F_3)$ 的共轭问题。
Limit pretrees for free group automorphisms: existence
本文为任意自由群自同构构造具有刚性作用、扩张自同构及双曲元素增长检测性质的实前树，推广度量树理论，并以前树语言实现等变爆破。
We present a normal form for outer automorphisms $ϕ$ of a non-abelian free group $F_N$ which grow quadratically (measured through the maximal growth of conjugacy classes in $F_N$ under iteration of $ϕ$). In analogy to the known normal form for linearly growing automorphisms as efficient Dehn twist, our normal form for $ϕ$ is given in terms of a 2-level Dehn twist on a graph-of-groups $\cal{G}$ with $π_1 {\cal{G}} \cong F_N$, where a conjugacy class of $F_N$ grows at most linearly if and only if it is contained in a vertex group of $\cal{G}$. Our proof is based on earlier work of the second aut…
Twisted Conjugacy Growth of the Generalised Heisenberg Groups
该研究确定了广义Heisenberg群自同构的扭曲共轭增长函数形式，通过上下中心序列的维数与秩区分两类增长形式，并证明扭曲共轭增长不超过共轭增长。
We study the automorphism groups of free-by-cyclic groups and show these are finitely generated in the following cases: (i) when defining automorphism has linear growth and (ii) when the rank of the underlying free group has rank at most 3. The techniques we use are actions on trees, including the trees of cylinders due to Guirardel and Levitt, the relative hyperbolicity of free-by-cyclic groups (due to Gautero and Lustig, Ghosh, and Dahmani and Li) and the filtration of the automorphisms of a group preserving a tree, following Bass and Jiang, and Levitt. Our general strategy is to produce an …
In the first paper of this series (arxiv.org/abs/1210.2961) we studied the asymptotic behavior of Betti numbers, twisted torsion and other spectral invariants for sequences of lattices in Lie groups G. A key element of our work was the study of invariant random subgroups (IRSs) of G. Any sequence of lattices has a subsequence converging to an IRS, and when G has higher rank, the Nevo-Stuck-Zimmer theorem classifies all IRSs of G. Using the classification, one can deduce asymptotic statments about spectral invariants of lattices. When G has real rank one, the space of IRSs is more complicated. …
Fixing a subgroup $Γ$ in a group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $Δ$ of $G$ with $[Γ: Γ\cap Δ][Δ: Γ\cap Δ] = n$. For pairs $Γ\leq A$, where $A$ is the automorphism group of a $p$-regular tree and $Γ$ is finitely generated, we show that this function can take on finite, countable, or uncountable cardinals. For almost all known branch groups $Γ$ (the first Grigorchuk group, the twisted twin Grigorchuk group, Pervova groups, Gupta-Sidki groups, etc.) acting on $p$-regular trees, this function is precisely $\aleph_0$ for any $n …
We prove that the growth rate of an endomorphism of a finitely generated nilpotent group equals to the growth rate of induced endomorphism on its abelinization, generalizing the corresponding result for an automorphism in [14]. We also study growth rates of endomorphisms for specific solvable groups, lattices of Sol, providing a counterexample to a known result in [5] and proving that the growth rate is an algebraic number.
Metric dimension and product entropy of group $C^{\ast}$-algebras
研究有限生成离散群的缩减C*代数的度量维数及群C*代数自同构的乘积熵，通过多项式增长性质给出熵的上下界，证明指数增长群的C*代数度量维数通常为无穷大。
Let $F$ be a free group of positive, finite rank and let $Φ\in Aut(F)$ be a polynomial-growth automorphism. Then $F\rtimes_Φ\mathbb Z$ is strongly thick of order $η$, where $η$ is the rate of polynomial growth of $ϕ$. This fact is implicit in work of Macura, but her work predates the notion of thickness. Therefore, in this note, we make the relationship between polynomial growth and thickness explicit. Our result combines with a result independently due to Dahmani-Li, Gautero-Lustig, and Ghosh to show that free-by-cyclic groups admit relatively hyperbolic structures with thick peripheral subgr…
In this paper we start the inquiry into proving uniform exponential growth in the context of groups acting on CAT(0) cube complexes. We address free group actions on CAT(0) square complexes and prove a more general statement. This says that if $F$ is a finite collection of hyperbolic automorphisms of a CAT(0) square complex $X$, then either there exists a pair of words of length at most 10 in $F$ which freely generate a free semigroup, or all elements of $F$ stabilize a flat (of dimension 1 or 2 in $X$). As a corollary, we obtain a lower bound for the growth constant, $\sqrt[10]{2}$, which is …
Mapping class groups of exotic tori and actions by ${\rm SL}_d(\mathbb Z)$
该研究确定了维数非4的奇异环面中保向微分同痕类群到SL_d(Z)的同态为分裂满射的情形，计算了一类奇异环面的映射类群，证明了其SL_d(Z)作用的同调性质，并得出许多奇异环面无非平凡SL_d(Z)微分作用的结论。
Let $ϕ:G \to G$ be a group endomorphism where $G$ is a finitely generated group of exponential growth, and denote by $R(ϕ)$ the number of twisted $ϕ$-conjugacy classes. Fel'shtyn and Hill \cite{fel-hill} conjectured that if $ϕ$ is injective, then $R(ϕ)$ is infinite. This conjecture is true for automorphisms of non-elementary Gromov hyperbolic groups, see \cite{ll} and \cite {fel:1}. It was showed in \cite {gw:2} that the conjecture does not hold in general. Nevertheless in this paper, we show that the conjecture holds for the Baumslag-Solitar groups $B(m,n)$, where either $|m|$ or $|n|$ is gre…
For an interval exchange map, the number of discontinuities of its iterates either exhibits linear growth or is bounded. This dichotomy is used to prove that the group of interval exchanges does not contain distortion elements, giving examples of groups that do not act faithfully via interval exchanges. As a further application of this dichotomy, a classification of centralizers in the group of interval exchanges is given. This classification of centralizers is used to compute the automorphism group of the interval exchange group.
This paper, which is the last of a series of three papers, studies dynamical properties of elements of $\mathrm{Out}(F_{\tt n})$, the outer automorphism group of a nonabelian free group $F_{\tt n}$. We prove that, for every subgroup $H$ of $\mathrm{Out}(F_{\tt n})$, there exists an element $ϕ\in H$ such that, for every element $g$ of $F_{\tt n}$, the conjugacy class $[g]$ has polynomial growth under iteration of $ϕ$ if and only if $[g]$ has polynomial growth under iteration of every element of $H$.
On polynomial free-by-cyclic groups
研究 free-by-cyclic 群中自由群自同构单值性的动力学性质在拟等距下的不变性，利用循环分裂和细长分裂证明并刻画了单值性增长类型作为几何不变量的特性。
Group cohomology of polynomial growth is defined for any finitely generated discrete group, using cochains that have polynomial growth with respect to the word length function. We give a geometric condition that guarantees that it agrees with the usual group cohomology and verify this condition for a class of combable groups. Our condition involves a chain complex that is closely related to exotic cohomology theories studied by Allcock and Gersten and by Mineyev.
For every infinite sequence $ω=x_1,x_2,...$, with $x_i\in\{0,1\}$, we construct an infinite 4-regular graph $X_ω$. These graphs are precisely the Schreier graphs of the action of a certain self-similar group on the space $\{0,1\}^{\infty}$. We solve the isomorphism and local isomorphism problems for these graphs, and determine their automorphism groups. Finally, we prove that all graphs $X_ω$ have intermediate growth.
For each prime p and a monic polynomial f, invertible over p, we define a group G_{p,f} of p-adic automorphisms of the p-ary rooted tree. The groups are modeled after the first Grigorchuk group, which in this setting is the group G_{2,x^2+x+1}. We show that the constructed groups are self-similar, regular branch groups. This enables us to calculate the Hausdorff dimension of their closures, providing concrete examples (not using random methods) of topologically finitely generated closed subgroups of the group of p-adic automorphisms with Hausdorff dimension arbitrarily close to 1. We provide a…
Structure of an exotic $2$-local subgroup in $E_7(q)$
研究$E_7(q)$中2-局部子群的结构，证明其在特定条件下与自同构群的关系，揭示其在有限简单李型群中的最大子群性质。
Exponential growth rates in hyperbolic groups (after Koji Fujiwara and Zlil Sela)
本文概述了 Fujiwara 和 Sela 的研究，证明非初等双曲群的指数增长率集合是良序的且序型至少为 $ω^ω$，每个增长率仅由有限多个生成集实现，建立了与双曲3-流形体积定理对应的群论结果。
We consider groups of automorphisms of locally finite trees, and give conditions on its subgroups that imply that they are not elementary amenable. This covers all known examples of groups that are not elementary amenable and act on the trees: groups of intermediate growths and Basilica group, by giving a more straightforward proof. Moreover, we deduce that all finitely generated branch groups are not elementary amenable, which was conjectured by Grigorchuk.
A locally compact group G is said to be Hermitian if every selfadjoint element of L^1(G) has real spectrum. Using Halmos' notion of capacity in Banach algebras and a result of Jenkins, Fountain, Ramsay and Williamson we will put a bound on the growth of Hermitian groups. In other words, we will show that if G has a subset that grows faster than a certain constant, then G cannot be Hermitian. Our result allows us to give new examples of non-Hermitian groups which could not tackled by the existing theory. The examples include certain infinite free Burnside groups, automorphism groups of trees, a…
Let p and l be two distinct prime numbers and let G be a group. We study the asymptotic behaviour of the mod-l Betti numbers in p-adic analytic towers of finite index subgroups. If X is a finite l-group of automorphisms of G, our main theorem allows to lift lower bounds for the mod-l cohomology growth in the fixed point group G^X to lower bounds for the growth in G. We give applications to S-arithmetic groups and we also obtain a similar result for cohomology with rational coefficients.
Real Rational Surface Automorphisms : Positivity and Linearity
本文研究保持尖点三次曲线的实有理曲面自同构，通过自由基本群中的正半群与本原转移矩阵证明其正拓扑熵，并在 m+n 为奇数时给出伪 Anosov 实现。
We associate a 2-complex to the following data: a presentation of a semigroup $S$ and a transitive action of $S$ on a set $V$ by partial transformations. The automorphism group of the action acts properly discontinuously on this 2-complex. A sufficient condition is given for the 2-complex to be simply connected. As a consequence we obtain simple topological proofs of results on presentations of Schützenberger groups. We also give a geometric proof that a finitely generated regular semigroup with finitely many idempotents has polynomial growth if and only if all its maximal subgroups are virtua…
On transitive permutation groups with exponential graph growth
本文研究传递置换群的图增长问题，证明当群存在非平凡块且其补集的逐点稳定子非平凡时，该群的图增长必为指数级，从而推广了多个已有结果。
Let $\mathcal{T}_{q+1}$ be the $(q+1)$-regular tree and let $G$ be a group of automorphisms acting transitively on the vertices and on the boundary of $\mathcal{T}_{q+1}$. We give an upper bound for the growth of cocycles with values in any unitary representation of the group $G$. This bound is optimal by projecting the Haagerup cocycle onto an appropriate subspace of $\ell^{2}(E)$. We also obtain a description of functions conditionally of negative type which are unbounded.
We consider the actions of (semi)groups on a locally compact group by automorphisms. We show the equivalence of distality and pointwise distality for the actions of a certain class of groups. We also show that a compactly generated locally compact group of polynomial growth has a compact normal subgroup $K$ such that $G/K$ is distal and the conjugacy action of $G$ on $K$ is ergodic; moreover, if $G$ itself is (pointwise) distal then $G$ is Lie projective. We prove a decomposition theorem for contraction groups of an automorphism under certain conditions. We give a necessary and sufficient cond…
Structures with not too fast unlabelled growth
研究一类增长速率受限的结构$\mathscr{S}$，通过自同构群对其进行完整分类，证明其直至双可定义性结构可数，在$(\mathbb{Q};<)$中一阶可解释且与有限有界齐次结构相互可定义，还证实其结构直至相互可定义性有有限多个一阶约简。
In this article we prove a conjecture of A. Lubotzky: let G = G_0(K), where K is a local field of characteristic p>5, G_0 is a simply connected, absolutely almost simple K-group of K-rank at least 2. We give the rate of growth of r_x(G) :={H< G| H a lattice in G; vol(G/H)<x}/~, where H_1\simH_2 if and only if there is an abstract automorphism t of G such that H_2 =t(H_1). We also study the rate of subgroup growth s_x(H) of any lattice H in G. As a result we show that these two functions have the same rate of growth which proves Lubotzky's conjecture. Along the way, we also study the rate of gr…
The rates of growth in a hyperbolic group
本文研究双曲群及其子群、子半群在所有有限生成集下的增长率集合，证明这些集合良序并确定极限群情形的序数为 $ω^ω$。
In this paper we define a way to get a bounded invertible automaton starting from a finite graph. It turns out that the corresponding automaton group is regular weakly branch over its commutator subgroup, contains a free semigroup on two elements and is amenable of exponential growth. We also highlight a connection between our construction and the right-angled Artin groups. We then study the Schreier graphs associated with the self-similar action of these automaton groups on the regular rooted tree. We explicitly determine their diameter and their automorphism group in the case where the initi…
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