D. V. Osin
The aim of this note is to show that weak relative hyperbolicity of a group relative to a subgroup (or relative hyperbolicity in the sense of Farb) does not imply any natural analogues of some well-known algebraic properties of ordinary hyperbolic groups. Our main tools are combination theorems for weakly relatively hyperbolic groups.
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We show that word hyperbolicity of automorphism groups of graph products $G_Γ$ and of Coxeter groups $W_Γ$ depends strongly on the shape of the defining graph $Γ$. We also characterized those $Aut(G_Γ)$ and $Aut(W_Γ)$ in terms of $Γ$ that are virtually free.
Given a finite rank free group $\mathbb{F}$ of $\mathsf{rank}(\mathbb{F})\geq 3$, we show that the mapping torus of $ϕ$ is (strongly) relatively hyperbolic if $ϕ$ is exponentially growing. We combine our result with the work of Button-Kropholler to answer a question asked by Minasyan-Osin regarding the acylindrical hyperbolicity of such free-by-cyclic extensions. As an application we construct new examples of free-by-free hyperbolic extensions where the elements of the quotient group are not necessarily fully irreducible. We also give a new proof of the Bridson-Groves quadratic isoperimetric i…
Relative free splitting and free factor complexes I: Hyperbolicity
研究自由群$F_n$及一般群$\Gamma$相对于自由因子系统选择的相对自由分裂复形和相对自由因子复形的大规模几何，证明其双曲性，证明还给出复形中粗传递拟测地线族信息。
Acylindrical hyperbolicity for Artin groups with a visual splitting
该研究建立了判定具有可视分裂的Artin群柱形双曲性的准则，提出抛物子群弱反正规性相关猜想并在多类Artin群中证明，同时建立了相关Artin群猜想间的联系。
We examine a graph $Γ$ encoding the intersection of hyperplane carriers in a CAT(0) cube complex $\widetilde X$. The main result is that $Γ$ is quasi-isometric to a tree. This implies that a group $G$ acting properly and cocompactly on $\widetilde X$ is weakly hyperbolic relative to the hyperplane stabilizers. Using disc diagram techniques and Wright's recent result on the aymptotic dimension of CAT(0) cube complexes, we give a generalization of a theorem of Bell and Dranishnikov on the finite asymptotic dimension of graphs of asymptotically finite-dimensional groups. More precisely, we prove …
We show how to derive hyperbolicity of the free factor complex of $F_N$ from the Handel-Mosher proof of hyperbolicity of the free splitting complex of $F_N$, thus obtaining an alternative proof of a theorem of Bestvina-Feighn. We also show that under the natural map $τ$ from the free splitting complex to free factor complex, a geodesic $[x,y]$ maps to a path that is uniformly Hausdorff-close to a geodesic $[τ(x),τ(y)]$ .
We prove that the group of outer automorphisms of the free Coxeter group $W_n$ is acylindrically hyperbolic in the sense of Osin. As an application, we observe that any CAT(0) space admitting a geometric action by Out($W_n$) must contain a rank-one geodesic. The theorem proceeds from expanding on a well-known relationship between Out($W_n$) and the outer automorphism group of free groups.
We develop the geometry of folding paths in Outer space and, as an application, prove that the complex of free factors of a free group of finite rank is hyperbolic.
Hyperbolicity of Multiple Ascending HNN Extensions of Free Groups
研究多重上升HNN扩张的超几何性，基于超几何非满射自同构的推广，分析其结构特性。
We prove that for a free product $G$ with free factor system $\mathcal{G}$, any automorphism $ϕ$ preserving $\mathcal{G}$, atoroidal (in a sense relative to $\mathcal{G}$) and none of whose power send two different conjugates of subgroups in $\mathcal{G}$ on conjugates of themselves by the same element, gives rise to a semidirect product $G\rtimes_ϕ\mathbb{Z}$ that is relatively hyperbolic with respect to suspensions of groups in $\mathcal{G}$. We recover a theorem of Gautero-Lustig and Ghosh that, if $G$ is a free group, $ϕ$ an automorphism of $G$, and $\mathcal{G}$ is its family of polynomia…
It is conjectured that the central quotient of every irreducible Artin group is either virtually cyclic or acylindrically hyperbolic. We prove this conjecture for Artin groups associated to triangle-free graphs and Artin groups of large type associated to cones over square-free bipartite graphs. In fact, we treat Artin groups that are known to be CAT(0) groups by a result of Brady and McCammond.
We prove that the free splitting complex of a finite rank free group, also known as Hatcher's sphere complex, is hyperbolic.
This paper considers the question of relative hyperbolicity of an Artin group with regard to the geometry of its associated Deligne complex. We prove that an Artin group is weakly hyperbolic relative to its finite (or spherical) type parabolic subgroups if and only if its Deligne complex is a Gromov hyperbolic space. For a 2-dimensional Artin group the Deligne complex is Gromov hyperbolic precisely when the corresponding Davis complex is Gromov hyperbolic, that is, precisely when the underlying Coxeter group is a hyperbolic group. For Artin groups of FC type we give a sufficient condition for …
A group obtained from a nontrivial group by adding one generator and one relator which is a proper power of a word in which the exponent-sum of the additional generator is one contains the free square of the initial group and almost always (with one obvious exception) contains a non-abelian free subgroup. If the initial group is involution-free or the relator is at least third power, then the obtained group is SQ-universal and relatively hyperbolic with respect to the initial group.
We introduce the notion of strictly systolic angled complexes. They generalize Januszkiewickz and Świątkowski's $7$-systolic simplicial complexes and also their metric counterparts, which appear as natural analogues to Huang and Osajda's metrically systolic simplicial complexes in the context of negative curvature. We prove that strictly systolic angled complexes and the groups that act on them geometrically, together with their finitely presented subgroups, are hyperbolic. We use these complexes to study the geometry of one-relator groups without torsion, and prove hyperbolicity of such group…
Acylindrical hyperbolicity and the centers of Artin groups that are not free of infinity
本文引入约化团-立方复形，证明了与非完全图相关的不可约Artin群的非柱面双曲性，并得出此类群中心有限且多为平凡的结论。
Around subgroups of Artin groups: derived subgroups and acylindrical hyperbolicity in the even FC-case
该研究将直角Artin群的多项相关结果推广至更广泛的Artin群类，涵盖导子群自由性与群凝聚性的等价性、有限生成正规子群性质及FC型偶Artin群子群的柱形双曲性三方面内容。
This paper studies the generic behavior of $k$-tuple elements for $k\ge 2$ in a proper group action with contracting elements, with applications towards relatively hyperbolic groups, CAT(0) groups and mapping class groups. For a class of statistically convex-cocompact action, we show that an exponential generic set of $k$ elements for any fixed $k\ge 2$ generates a quasi-isometrically embedded free subgroup of rank $k$. For $k=2$, we study the sprawl property of group actions and establish that the class of statistically convex-cocompact actions is statistically hyperbolic in a sense of M. Duc…
Relative hyperbolicity of ascending HNN extension of groups
该研究证明了带自由因子系统的有限生成群的保系统单自同态对应的升序HNN扩张的相对双曲性，并得出有限秩自由群指数增长单自同态的升序HNN扩张相对双曲的推论，推进了几何群论相关研究。
Relative hyperbolicity of hyperbolic-by-cyclic groups
本文证明了无挠双曲群的自同构映射环面相对于极大多项式增长子群的悬挂具有相对双曲性，并给出了相应的典范子群集合。
Relative hyperbolicity of free extensions of free groups
本文给出了自由乘自由群具有保持尖点结构的相对双曲性的充要条件，证明了当一族指数增长外自同构共享不变子群系统且补集中共轭类均指数增长时，可由该系统构造外围子群使相应扩张相对双曲。
This article studies the properties of word-hyperbolic semigroups and monoids, i.e. those having context-free multiplication tables with respect to a regular combing, as defined by Duncan & Gilman. In particular, the preservation of word-hyperbolicity under taking free products is considered. Under mild conditions on the semigroups involved, satisfied e.g. by monoids or regular semigroups, we prove that the semigroup free product of two word-hyperbolic semigroups is again word-hyperbolic. Analogously, with a mild condition on the uniqueness of representation for the identity element, satisfied…
Handel and Mosher have proved that the free splitting complex FS for the free group is Gromov hyperbolic. This is a deep and much sought-after result, since it establishes FS as a good analogue of the curve complex for surfaces. We give a shorter alternative proof of this theorem, using surgery paths in Hatcher's sphere complex (another model for the free splitting complex), instead of Handel and Mosher's fold paths. As a byproduct, we get that surgery paths are unparameterized quasi-geodesics in the sphere complex. We explain how to deduce from our proof the hyperbolicity of some other comple…
We propose the metric notion of strong hyperbolicity as a way of obtaining hyperbolicity with sharp additional properties. Specifically, strongly hyperbolic spaces are Gromov hyperbolic spaces that are metrically well-behaved at infinity, and, under weak geodesic assumptions, they are strongly bolic as well. We show that CAT(-1) spaces are strongly hyperbolic. On the way, we determine the best constant of hyperbolicity for the standard hyperbolic plane. We also show that the Green metric defined by a random walk on a hyperbolic group is strongly hyperbolic. A measure-theoretic consequence at t…
We prove that if $G$ is a free-torsion group and $w(t)$ is a word in the alphabet $G \sqcup \{t^{\pm 1}\}$ with exponent sum one, then the group $<G,t|(w(t))^k = 1>$, where $k \geq 2$, is relatively hyperbolic with respect to $G$.
In this note, we explore the notion of hyperbolicity of topologically finitely generated profinite groups. Some applications to diophantine geometry are suggested and we try to reformulate certain problems in diophantine geometry in terms of hyperbolic profinite groups. Then, we introduce many occasions in which Galois groups are free profinite and try to explore implications of this condition in the world of diophantine geometry. In particular, we prove that, Grothendieck's "section conjecture" plus Shafarevich's "freeness conjecture" imply that hyperbolic curves have infinitely many solution…
Submanifold projections and hyperbolicity in ${\rm Out}(F_n)$
本文利用自由分裂图的几何实现构造子流形投影，并据此为 $n\geq 3$ 构建了一个新的双曲 ${\rm Out}(F_n)$-图，证明了 $n=3$ 时指数增长元素具有正平移长度。
We prove that the group STame($k^3$) of special tame automorphisms of the affine 3-space is not simple, over any base field of characteristic zero. Our proof is based on the study of the geometry of a 2-dimensional simply-connected simplicial complex C on which the tame automorphism group acts naturally. We prove that C is contractible and Gromov-hyperbolic, and we prove that Tame($k^3$) is acylindrically hyperbolic by finding explicit loxodromic weakly proper discontinuous elements.
Hyperbolicity and obstructions to PL actions of the circle
该研究探讨作用于圆周的分段线性同胚的 acylindrically 双曲群的结构，证明其为虚自由群等三类，推广直角 Artin 子群性质，还得出高维双曲流形基本群无法作用于圆周的结论。
We prove that every countable acylindrically hyperbolic group admits a highly transitive action with finite kernel. This theorem uniformly generalizes many previously known results and allows us to answer a question of Garion and Glassner on the existence of highly transitive faithful actions of mapping class groups. It also implies that in various geometric and algebraic settings, the transitivity degree of an infinite group can only take two values, namely $1$ and $\infty$. Here by transitivity degree of a group we mean the supremum of transitivity degrees of its faithful permutation represe…
Spaces that can be ordered effectively: virtually free groups and hyperbolicity
本文基于旅行商问题研究度量空间的渐近不变量，通过排序效率对群和空间进行分类，证明了 virtually free groups 可由 $4$ 点子集上的排序效率刻画，且所有 $δ$-hyperbolic spaces 在点数趋于无穷时可被极其高效地排序。
We prove a combination theorem for hyperbolic groups, in the case of groups acting on complexes displaying combinatorial features reminiscent of non-positive curvature. Such complexes include for instance weakly systolic complexes and C'(1/6) small cancellation polygonal complexes. Our proof involves constructing a potential Gromov boundary for the resulting groups and analyzing the dynamics of the action on the boundary in order to use Bowditch's characterization of hyperbolicity. A key ingredient is the introduction of a combinatorial property that implies a weak form of non-positive curvatu…
This article is part of the program of studying large-scale geometric properties of totally disconnected locally compact groups, TDLC-groups, by analogy with the theory for discrete groups. We provide a characterization of hyperbolic TDLC-groups, in terms of homological isoperimetric inequalities. This characterization is used to prove the main result of the article: for hyperbolic TDLC-groups with rational discrete cohomological dimension $\leq 2$, hyperbolicity is inherited by compactly presented closed subgroups. As a consequence, every compactly presented closed subgroup of the automorphis…
The question which motivates the article is the following: given a group acting on a CAT(0) cube complex, how can we prove that it is acylindrically hyperbolic? Keeping this goal in mind, we show a weak acylindricity of the action on the contact graph associated to a CAT(0) cube complex, and we prove a characterisation of WPD contracting isometries of CAT(0) cube complexes. As a first consequence, we find alternative arguments to show several criteria which were proved by Indira Chatterji and Alexandre Martin. Next, we show that if a group acts essentially and acylindrically on the hyperplanes…
We prove that the automorphism group of every infinitely-ended finitely generated group is acylindrically hyperbolic. In particular $\mathrm{Aut}(\mathbb{F}_n)$ is acylindrically hyperbolic for every $n\ge 2$. More generally, if $G$ is a group which is not virtually cyclic, and hyperbolic relative to a finite collection $\mathcal{P}$ of finitely generated proper subgroups, then $\mathrm{Aut}(G,\mathcal{P})$ is acylindrically hyperbolic. As a consequence, a free-by-cyclic group $\mathbb{F}_n\rtimes_φ\mathbb{Z}$ is acylindrically hyperbolic if and only if $φ$ has infinite order in $\mathrm{Out}(…
Polynomial hyperbolicity and products of free groups
本文研究了局部有限图在某种条件下成为多项式超几何空间的性质，证明了在紧致特殊群中，不含自由群F₂×F₂作为子群等价于是线性多项式超几何的，且包含F₂×F₂作为子群在紧致特殊群中是准等距不变的。
Spectral Dehn functions and a characterisation of word-hyperbolicity
本文引入谱德恩函数，证明其与德恩函数的关系，并通过谱德恩函数的度量自由变体刻画词超几何性，同时展示其比德恩函数提供更细致的信息。
We study the geometry of nonrelatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with nonrelatively hyperbolic peripheral subgroups is a quasi-isometry invariant. As an application, Artin groups are relatively hyperbolic if and only if freely decomposable. We also introduce a new quasi-isometry invariant of metric spaces called metrically thick, which is sufficien…
Building on previous results concerning hyperbolicity of groups of Fibonacci type, we give an almost complete classification of the (non-elementary) hyperbolic groups within this class. We are unable to determine the hyperbolicity status of precisely two groups, namely the Gilbert-Howie groups H(9,4), H(9,7). We show that if H(9,4) is torsion-free then it is not hyperbolic. We consider the class of T(5) cyclically presented groups and classify the (non-elementary) hyperbolic groups and show that the Tits alternative holds.
Let X be a proper CAT(0) cube complex admitting a proper cocompact action by a group G. We give three conditions on the action, any one of which ensures that X has a factor system in the sense of [BHS14]. We also prove that one of these conditions is necessary. This combines with results of Behrstock--Hagen--Sisto to show that $G$ is a hierarchically hyperbolic group; this partially answers questions raised by those authors. Under any of these conditions, our results also affirm a conjecture of BehrstockHagen on boundaries of cube complexes, which implies that X cannot contain a convex stairca…
The main result in this paper is the failure of the finitely generated intersection property (FGIP) of ascending HNN extensions of non-cyclic finite rank free groups. This class of group consists of free-by-cyclic groups and properly ascending HNN extensions of free groups. We also give a sufficient condition for the failure of the FGIP in the context of relative hyperbolicity, we apply this to free-by-cyclic groups of exponential growth.
For a given divison algebra of the quaternions we construct two types of units: Pell units and Gauss units. If K is a rational quadratic extension and G is a finite group, we classify R and G, s.t., the unit group U(RG) of augmentation one is hyperbolic. In particular we give necessary and sufficient conditions when G is the quaternion group of order 8. In this case, the hyperbolic boundary is isomorphic to the 2-dimensional sphere. As a result we reach a class of groups of one end, that are not virtually free.
In this paper, we introduce a geometric statistic called the "sprawl" of a group with respect to a generating set, based on the average distance in the word metric between pairs of words of equal length. The sprawl quantifies a certain obstruction to hyperbolicity. Group presentations with maximum sprawl (i.e., without this obstruction) are called statistically hyperbolic. We first relate sprawl to curvature and show that nonelementary hyperbolic groups are statistically hyperbolic, then give some results for products, for Diestel-Leader graphs and lamplighter groups. In free abelian groups, t…
In this paper we have examined hyperbolicity and relative hyperbolicity of $Γ_n := \mathsf{Out}(G_n)$ , where $G_n = A_1*...*A_n$, is a finite free product and each $A_i$ is a finite group. We have used the $\mathsf{Out}(G_n)$ action on the Guirardel-Levitt deformation space, to find a virtual generating set and prove quasi isometric embedding of a large class of subgroups. We have used ideas from works of Mosher-Handel and Alibegović to prove non-distortion. We have used these subgroups to prove that $Γ_n$ is thick for higher complexities. Thickness was developed by Behrstock-Druţu-Mosher and…
Two applications of strong hyperbolicity
本文利用双曲群可配备强双曲度量的事实，给出了两个解析应用：构造交叉积C*-代数上的自然时间流及其唯一KMS态，并简短证明了双曲群在足够大的$\u2113^p$空间上有真等距作用。
Cyclic hyperbolicity in CAT(0) cube complexes
本文证明余紧特殊群不含F2×F2当且仅当循环双曲，并在CAT(0)立方复形几何作用群中给出结构定理和强Tits替代，推论SQ-普遍性。
Four geometric conditions on a geodesic metric space, which are stronger variants of classical conditions characterizing hyperbolicity, are proved to be equivalent. In the particular case of the Cayley graph of a finitely generated group, it is shown that they characterize virtually free groups.
Ghys and Sergiescu proved in the $80$s that Thompson's group $T$, and hence $F$, admits actions by $C^{\infty}$ diffeomorphisms of the circle . They proved that the standard actions of these groups are topologically conjugate to a group of $C^\infty$ diffeomorphisms. Monod defined a family of groups of piecewise projective homeomorphisms, and Lodha-Moore defined finitely presentable groups of piecewise projective homeomorphisms. These groups are of particular interest because they are nonamenable and contain no free subgroup. In contrast to the result of Ghys-Sergiescu, we prove that the group…
On free components of Artin and Coxeter groups
研究Artin和Coxeter群的连通分量数量，探讨von Neumann代数与自由群因子问题的关系，以及测度等价中的相关情况。
We analyze the structure of the \emph{frequency space} $Q(F)$ of a nonabelian free group $F=F(a_1,...,a_k)$ consisting of all shift-invariant Borel probability measures on $\partial F$ and construct a natural action of $Out(F)$ on $Q(F)$. In particular we prove that for any outer automorphism $ϕ$ of $F$ the \emph{conjugacy distortion spectrum} of $ϕ$, consisting of all numbers $||ϕ(w)||/||w||$, where $w$ is a nontrivial conjugacy class, is the intersection of $\mathbb Q$ and a closed subinterval of $\mathbb R$ with rational endpoints. We also provide an algorithm for detecting strict hyperboli…
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