Anna Dioubina
We construct examples of groups showing that virtual solvability and the property of being virtually torsion-free are not preserved by bi-Lipschitz maps and hence by quasi-isometries.
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We study the quasi-isometric rigidity of a large family of finitely generated groups that split as graphs of groups with virtually free vertex groups and two-ended edge groups. Let $G$ be a group that is one-ended, hyperbolic relative to virtually abelian subgroups, and has JSJ decomposition over two-ended subgroups containing only virtually free vertex groups that aren't quadratically hanging. Our main result is that any group quasi-isometric to $G$ is abstractly commensurable to $G$. In particular, our result applies to certain "generic" HNN extensions of a free group over cyclic subgroups.
Robust quasi-isometric embeddings of virtually free groups
论文研究了非元素自由群在GL_n(k)中的稳健准等距嵌入，展示了非Anosov表示的示例，并探讨了自由半群的嵌入特性。
In 1985, Dunwoody showed that finitely presentable groups are accessible. Dunwoody's result was used to show that context-free groups, groups quasi-isometric to trees or finitely presentable groups of asymptotic dimension 1 are virtually free. Using another theorem of Dunwoody of 1979, we study when a group is virtually free in terms of its Cayley graph and we obtain new proofs of the mentioned results and other previously depending on them.
We give an algorithm for solving equations and inequations with rational constraints in virtually free groups. Our algorithm is based on Rips classification of measured band complexes. Using canonical representatives, we deduce an algorithm for solving equations and inequations in hyperbolic groups (maybe with torsion). Additionnally, we can deal with quasi-isometrically embeddable rational constraints.
Automorphism groups of compact complex surfaces: T-Jordan property, Tits alternative and solvability
本研究针对紧复曲面的双解析自同构群，证明其挠子群具有几乎幂零性，并探究了该群的Tits二择性与几乎可解子群的几乎导出长度，推进了复曲面自同构群结构的相关研究。
Two groups are virtually isomorphic if they can be obtained one from the other via a finite number of steps, where each step consists in taking a finite extension or a finite index subgroup (or viceversa). Virtually isomorphic groups are always quasi-isometric, and a group G is quasi-isometrically rigid if every group quasi-isometric to G is virtually isomorphic to G. In this survey we describe quasi-isometric rigidity results for fundamental groups of manifolds which can be decomposed into geometric pieces. After stating by now classical results on lattices in semisimple Lie groups, we focus …
Quasi-actions whose quasi-orbits are quasi-isometric to trees
本文给出了任意群在度量空间上的拟作用可归约为有界度数树上有界等距作用的充要条件，并证明若拟作用度量正则且拟轨道与树拟等距，则该群是虚拟自由的。
We prove that if $T$ is an $\mathbb R$-tree with a minimal free isometric action of $F_N$, then the $Out(F_N)$-stabilizer of the projective class $[T]$ is virtually cyclic. For the special case where $T=T_+(ϕ)$ is the forward limit tree of an atoroidal iwip element $ϕ\in Out(F_N)$ this is a consequence of the results of Bestvina, Feighn and Handel, via very different methods. We also derive a new proof of the Tits alternative for subgroups of $Out(F_N)$ containing an iwip (not necessarily atoroidal): we prove that every such subgroup $G\le Out(F_N)$ is either virtually cyclic or contains a fre…
Failure of quasi-isometric rigidity for infinite-ended groups
本研究证明了满足特定条件的无限端群不具备拟等距刚性，结合已有工作得出剩余有限多端双曲群拟等距刚性的充要条件，证明方法改编自Whyte的相关论证。
Generalised Baumslag-Solitar groups and Hierarchically Hyperbolic Groups
本文研究任意维数广义Baumslag-Solitar群在双曲空间上的等距作用，证明实质上成为分层双曲群及性质（QT）在拟等距下均非不变量。
A subgroup $H\leq G$ is said to be almost normal if every conjugate of $H$ is commensurable to $H$. If $H$ is almost normal, there is a well-defined quotient space $G/H$. We show that if a group $G$ has type $F_{n+1}$ and contains an almost normal coarse $PD_n$ subgroup $H$ with $e(G/H)=\infty$, then whenever $G'$ is quasi-isometric to $G$, it contains an almost normal subgroup $H'$ that is quasi-isometric to $H$. Moreover, the quotient spaces $G/H$ and $G'/H'$ are quasi-isometric. This generalises a theorem of Mosher-Sageev-Whyte, who prove the case in which $G/H$ is quasi-isometric to a fini…
Quasisymmetric universality, quasi-isometric classification and topological rigidity of Kleinian groups
该研究针对克莱因群极限集，通过刻画几何有限极限集的拟对称普适性，实现有限生成克莱因群的拟等距分类及几乎拓扑刚性几何有限克莱因群的分类
Accessibility, planar graphs, and quasi-isometries
本文证明了可及的平面图与准等距图之间的关系，揭示了有限生成群在准等距下的分类，指出此类群本质上是自由积的虚拟自由群或曲面群。
The hyperbolic plane admits a quasi-isometric embedding into a hyperbolic group if and only if the group is not virtually free.
Quasi-isometric rigidity of the integers: an elementary primer
本文提供了一个初等证明，展示任何有限生成且准等距于整数的群都是整数的虚群。
Property (VRC) and virtual fibering for amalgamated free products
本文研究了 amalgamated 自由积的性质，证明了在特定条件下该结构具有虚拟阿贝尔群的性质，并探讨了其残余性质和纤维化条件。
We investigate the structure of the minimal displacement set in $8$-located complexes with the SD'-property. We show that such set embeds isometrically into the complex. Since $8$-location and simple connectivity imply Gromov hyperbolicity, the minimal displacement set in such complex is systolic. Using these results, we construct a low-dimensional classifying space for the family of virtually cyclic subgroups of a group acting properly on an $8$-located complex with the SD'-property.
Given a bounded valence, bushy tree T, we prove that any cobounded quasi-action of a group G on T is quasiconjugate to an action of G on another bounded valence, bushy tree T'. This theorem has many applications: quasi-isometric rigidity for fundamental groups of finite, bushy graphs of coarse PD(n) groups for each fixed n; a generalization to actions on Cantor sets of Sullivan's Theorem about uniformly quasiconformal actions on the 2-sphere; and a characterization of locally compact topological groups which contain a virtually free group as a cocompact lattice. Finally, we give the first exam…
Sela introduced limit groups in his work on the Tarski problem, and showed that each limit group has a cyclic hierarchy. In this paper, a class of relatively hyperbolic groups, equipped with a hierarchy similar to the one for limit groups, is shown to be closed under quasi-isometry. Additionally, these groups share some of the properties of limit groups. In particular, groups quasi-isometric to limit groups are shown to be LERF and virtually toral relatively hyperbolic.
Let G be a group acting isometrically with discrete orbits on a separable complete CAT(0)-space of bounded topological dimension. Under certain conditions, we give upper bounds for the Bredon cohomological dimension of G for the families of finite and virtually cyclic subgroups. As an application, we prove that the mapping class group of any closed, connected, and orientable surface of genus g greater than 1 admits a (9g-8)-dimensional classifying space with virtually cyclic stabilizers. In addition, our results apply to fundamental groups of graphs of groups and groups acting on Euclidean bui…
A monoid is said to be special if it admits a presentation in which all defining relations are of the form $w = 1$. Groups are familiar examples of special monoids. This article studies the geometric and structural properties of the Cayley graphs of finitely presented special monoids, building on work by Zhang and Gray-Steinberg. It is shown that the right Cayley graph $Γ$ of a special monoid $M$ is a context-free graph, in the sense of Muller & Schupp, if and only if the group of units of $M$ is virtually free. This generalises the geometric aspect of the well-known Muller-Schupp Theorem from…
We consider a finitely generated virtually abelian group $G$ acting properly and without inversions on a CAT(0) cube complex $X$. We prove that $G$ stabilizes a finite dimensional CAT(0) subcomplex $Y \subseteq X$ that is isometrically embedded in the combinatorial metric. Moreover, we show that $Y$ is a product of finitely many quasilines. The result represents a higher dimensional generalization of Haglund's axis theorem.
A nonpolycyclic nilpotent-by-cyclic group Gamma can be expressed as the HNN extension of a finitely-generated nilpotent group N. The first main result is that quasi-isometric nilpotent-by-cyclic groups are HNN extensions of quasi-isometric nilpotent groups. The nonsurjective injection defining such an extension induces an injective endomorphism phi of the Lie algebra g associated to the Lie group in which N is a lattice. A normal form for automorphisms of nilpotent Lie algebras--permuted absolute Jordan form-- is defined and conjectured to be a quasi-isometry invariant. We show that if phi, th…
It is well-known that a complete Riemannian manifold M which is locally isometric to a symmetric space is covered by a symmetric space. Here we prove that a discrete version of this property (called local to global rigidity) holds for a large class of vertex-transitive graphs, including Cayley graphs of torsion-free lattices in simple Lie groups, and Cayley graph of torsion-free virtually nilpotent groups. By contrast, we exhibit various examples of Cayley graphs of finitely presented groups (e.g. SL(4,Z)) which fail to have this property, answering a question of Benjamini, Ellis, and Georgako…
We prove that the class of convex-cocompact Kleinian groups is quasi-isometrically rigid. We also establish that a word hyperbolic group with a planar boundary different from the sphere is virtually a convex-cocompact Kleinian group provided that its boundary has Ahlfors regular conformal dimension strictly less than $2$ or if it acts geometrically on a CAT(0) cube complex.
In this note, we announce the first results on quasi-isometric rigidity of non-nilpotent polycyclic groups. In particular, we prove that any group quasi-isometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We prove analogous results for groups quasi-isometric to $R \ltimes R^n$ where the semidirect product is defined by a diagonalizable matrix of determinant one with no eigenvalues on the unit circle. Our approach to these problems is to first classify all self quasi-isometries of the solvable Lie group. Our classification of self quasi-isometries for $R …
We study the large scale geometry of mapping class groups MCG(S), using hyperbolicity properties of curve complexes. We show that any self quasi-isometry of MCG(S) (outside a few sporadic cases) is a bounded distance away from a left-multiplication, and as a consequence obtain quasi-isometric rigidity for MCG(S), namely that groups quasi-isometric to MCG(S) are virtually equal to it. (The latter theorem was proved by Hamenstadt using different methods). As part of our approach we obtain several other structural results: a description of the tree-graded structure on the asymptotic cone of MCG(S…
Subsets of groups with context-free preimages
本文研究有限生成群中可识别上下文无关子集，推广Muller-Schupp定理，证明有限可识别上下文无关子集的存在性、所有共轭类可识别上下文无关性以及陪集可识别上下文无关性均与群的几乎自由性或Schreier陪集图拟等距于树等价。
String graphs are quasi-isometric to planar graphs
研究证明字符串图与平面图在准等距意义上等价，并展示了相关结论如Assouad-Nagata维数及群的结构，同时给出多项时间算法生成准等距平面图。
Characterizing hierarchically hyperbolic free by cyclic groups
本文通过研究具有粗中位的自由群，证明其等价于层次超几何群和CAT(0)立方复形的准等距性，提出'无分支块'条件和'过度线性'性质。
Lattice envelopes of right-angled Artin groups
本文构造非联图情形下直角 Artin 群的局部紧余紧格包络，并证明其半直积图自同构群为泛格包络，还刻画相关 Cayley 图全自同构群。
We exhibit a family of infinite, finitely-presented, nilpotent-by-abelian groups. Each member of this family is a solvable S-arithmetic group that is related to Baumslag-Solitar groups, and everyone of these groups has a quasi-isometry group that is virtually a product of a solvable real Lie group and a solvable p-adic Lie group. In addition, we propose a candidate for a polycyclic group whose quasi-isometry group is a solvable real Lie group, and we introduce a candidate for a quasi-isometrically rigid solvable group that is not finitely presented. We also record some conjectures on the large…
We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not admit peripheral splittings, contains a quasi-isometrically embedded copy of the hyperbolic plane. In natural situations, the specific embeddings we find remain quasi-isometric embeddings when composed with the inclusion map from the Cayley graph to the coned-off graph, as well as when composed with the quotient map to "almost every" peripheral (Dehn) filling. We apply our theorem to study the same question for fundamental groups of 3-manifolds. The key idea is to study quantitative geometric prop…
Generalizations of the Muller-Schupp theorem and tree-like inverse graphs
本文推广了Muller-Schupp定理，证明了拟传递逆图Γ拟等距于树、上下文无关及Aut(Γ)虚自由的等价性，并引入群论类比解决T. Brough猜想弱版本，刻画了自由群直积的有限生成子群的字问题。
Graphically discrete groups and rigidity
本文引入群的图离散性概念，证明顶点群为图离散群的群图具强刚性，包括自由积的作用刚性与特定双曲群图的拟等距刚性，还给出例子并指出其非可公度不变性。
We prove that if a quasi-isometry of warped cones is induced by a map between the base spaces of the cones, the actions must be conjugate by this map. The converse is false in general, conjugacy of actions is not sufficient for quasi-isometry of the respective warped cones. For a general quasi-isometry of warped cones, using the asymptotically faithful covering constructed in a previous work with Jianchao Wu, we deduce that the two groups are quasi-isometric after taking Cartesian products with suitable powers of the integers. Secondly, we characterise geometric properties of a group (coarse e…
Given a finitely generated subgroup $Γ\le \mathrm{Out}(\mathbb{F})$ of the outer automorphism group of the rank $r$ free group $\mathbb{F} = F_r$, there is a corresponding free group extension $1 \to \mathbb{F} \to E_Γ \to Γ\to 1$. We give sufficient conditions for when the extension $E_Γ$ is hyperbolic. In particular, we show that if all infinite order elements of $Γ$ are atoroidal and the action of $Γ$ on the free factor complex of $\mathbb{F}$ has a quasi-isometric orbit map, then $E_Γ$ is hyperbolic. As an application, we produce examples of hyperbolic $\mathbb{F}$-extensions $E_Γ$ for whi…
Two groups have a common model geometry if they act properly and cocompactly by isometries on the same proper geodesic metric space. The Milnor-Schwarz lemma implies that groups with a common model geometry are quasi-isometric; however, the converse is false in general. We consider free products of uniform lattices in isometry groups of rank-1 symmetric spaces and prove, within each quasi-isometry class, residually finite groups that have a common model geometry are abstractly commensurable. Our result gives the first examples of hyperbolic groups that are quasi-isometric but do not virtually …
Medvedev degrees of subshifts on groups
研究子移位在群上的梅德维埃德度数，通过代数和几何关系转移度数，对有限生成群的有限类型子移位进行分类，证明超几何平面准等距群具有非零度数。
Conformal dimension bounds, Pontryagin sphere boundaries, and algebraic fibering of right-angled Coxeter groups
研究直角考克斯特群，通过引入\((n,m)\) - 分支条件构造组合圆树并嵌入戴维斯复形，给出双曲群边界共形维数下界，展示相关图族，证明嵌入结果，得出群的共形维数等结论，升级相关工作。
Vertex-transitive graphs with uniformly bisecting quasi-geodesics
本文研究满足一致平分拟测地线性质的顶点可递图，证明其拟等距于欧氏或双曲平面，对应群为虚曲面群，正面解决了Kourovka笔记本的一个问题。
Twisted Equations and Palindrome Length
本文提出二次扭曲方程的初等算法，将自由群二次方程算法扩展至扭曲场景与有理约束，应用中证实回文长度不超过n的自由约化字的Catalan结构猜想，证明其为上下文无关语言且回文长度可高效计算。
Stable subgroups and the Morse boundary are two systematic approaches to collect and study the hyperbolic aspects of finitely generated groups. In this paper we unify and generalize these strategies by viewing any geodesic metric space as a countable union of stable subspaces: we show that every stable subgroup is a quasi--convex subset of a set in this collection and that the Morse boundary is recovered as the direct limit of the usual Gromov boundaries of these hyperbolic subspaces. We use this approach, together with results of Leininger--Schleimer, to deduce that there is no purely geometr…
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