G. Arzhantseva, A. Minasyan, D. Osin
We study residual properties of relatively hyperbolic groups. In particular, we show that if a group $G$ is non-elementary and hyperbolic relative to a collection of proper subgroups, then $G$ is SQ-universal.
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We show, using acylindrical hyperbolicity, that a finitely generated group splitting over $\Z$ cannot be simple. We also obtain SQ-universality in most cases, for instance a balanced group (one where if two powers of an infinite order element are conjugate then they are equal or inverse) which is finitely generated and splits over $\Z$ must either be SQ-universal or it is one of exactly seven virtually abelian exceptions.
In this article we produce an example of a non-residually finite group which admits a uniformly proper action on a Gromov hyperbolic space.
For a fixed word hyperbolic group we compare different residual properties related to quasiconvex subgroups.
We exhibit the first examples of residually finite non-linear Gromov hyperbolic groups. Our examples are constructed as amalgamated products of torsion-free cocompact lattices in the rank 1 Lie group $\mathrm{Sp}(d,1)$, $d\geq 2$ along maximal cyclic subgroups.
We classify finitely generated, residually finite automorphism-induced HNN-extensions in terms of the residual separability of a single associated subgroup. This classification provides a method to construct automorphism-induced HNN-extensions which are not residually finite. We prove that this method can never yield a "new" counter-example to Gromov's conjecture on the residual finiteness of hyperbolic groups.
Let $m$ and $k$ be integers such that $|m|, \, |k| >1$ and $\gcd (m,k)=1$. We show that all Baumslag-Solitar groups $BS(m,mk)$ are non-residually finite groups hyperbolic relative to residually finite subgroups. By a result of Osin (2007), this implies that there exists a non-residually finite hyperbolic group, thus solving a long-standing open problem of Gromov (1987). We also show that all $BS(m, mk)$ are non-Hopfian groups hyperbolic relative to Hopfian subgroups.
If there is a non-residually finite hyperbolic group, then there is a non-residually finite rigid hyperbolic group
该研究证明，一旦存在非剩余有限双曲群，便可构造出同样非剩余有限的刚性双曲群，从而将问题归结到刚性情形。
For any torsion-free hyperbolic group $Γ$ and any group $G$ that is fully residually $Γ$, we construct algorithmically a finite collection of homomorphisms from $G$ to groups obtained from $Γ$ by extensions of centralizers, at least one of which is injective. When $G$ is residually $Γ$, this gives a effective embedding of $G$ into a direct product of such groups. We also give an algorithmic construction of a diagram encoding the set of homomorphisms from a given finitely presented group to $Γ$.
We show that Out(G) is residually finite if G is a one-ended group that is hyperbolic relative to virtually polycyclic subgroups. More generally, if G is one-ended and hyperbolic relative to proper residually finite subgroups, the group of outer automorphisms preserving the peripheral structure is residually finite. We also show that Out(G) is virtually p-residually finite for every prime p if G is one-ended and toral relatively hyperbolic, or infinitely-ended and virtually p-residually finite.
We prove that if every hyperbolic group is residually finite, then every quasi-convex subgroup of every hyperbolic group is separable. The main tool is relatively hyperbolic Dehn filling.
We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable relatively hyperbolic group is residually finite. As a direct consequence, we obtain that the outer automorphism group of a limit group is residually finite.
Let $p$ be a prime. In this paper, we classify the geometric 3-manifolds whose fundamental groups are virtually residually $p$. Let $M=M^3$ be a virtually fibered 3-manifold. It is well-known that $G=π_1(M)$ is residually solvable and even residually finite solvable. We prove that $G$ is always virtually residually $p$. Using recent work of Wise, we prove that every hyperbolic 3-manifold is either closed or virtually fibered and hence has a virtually residually $p$ fundamental group. We give some generalizations to pro-$p$ completions of groups, mapping class groups, residually torsion-free ni…
The congruence subgroup property for mapping class groups and the residual finiteness of hyperbolic groups
本文在假设所有双曲群剩余有限的前提下，证明了有限型双曲曲面映射类群的同余子群性质，并由此推出profinite等价的双曲3-流形必为commensurable。
We give a quantification of residual finiteness for the fundamental groups of hyperbolic manifolds that admit a totally geodesic immersion to a compact, right-angled Coxeter orbifold of dimension 3 or 4. Specifically, we give explicit upper bounds on residual finiteness that are linear in terms of geodesic length. We then extend the linear upper bounds to hyperbolic manifolds with a finite cover that admits such an immersion. Since the quantifications are given in terms of geodesic length, we define the geodesic residual finiteness growth and show that this growth is equivalent to the usual re…
$H$ is called a $G$-subgroup of a hyperbolic group $G$ if for any finite subset $M\subset G$ there exists a homomorphism from $G$ onto a non-elementary hyperbolic group $G_1$ that is surjective on $H$ and injective on $M$. In his paper in 1993 A. Ol'shanskii gave a description of all $G$-subgroups in any given non-elementary hyperbolic group $G$. Here we show that for the same class of $G$-subgroups the finiteness assumption on $M$ (under certain natural conditions) can be replaced by an assumption of quasiconvexity.
We consider the class $\mathcal{R}$ of finitely generated toral relatively hyperbolic groups. We show that groups from $\mathcal{R}$ are commutative transitive and generalize a theorem proved by Benjamin Baumslag to this class. We also discuss two definitions of (fully) residually-$\mathcal{C}$ groups and prove the equivalence of the two definitions for $\mathcal{C}=\mathcal{R}$. This is a generalization of the similar result obtained by Ol'shanskii for $\mathcal{C}$ being the class of torsion-free hyperbolic groups. Let $Γ\in\mathcal{R}$ be non-abelian and non-elementary. We prove that every …
We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable hyperbolic group is residually finite. As a result, we are able to prove that the group of outer automorphisms of every finitely generated Fuchsian group and of every free-by-finite group is resudually finite. We also generalize the main result for relatively hyperbolic groups.
We give an effective upper bound, for certain arithmetic hyperbolic 3-manifold groups obtained from a quadratic form construction, on the minimal index of a subgroup that embeds in a fixed 6-dimensional right-angled reflection group, stabilizing a totally geodesic subspace. In particular, for manifold groups in any fixed commensurability class we show that the index of such a subgroup is asymptotically smaller than any fractional power of the volume of the manifold. We also give effective bounds on the geodesic residual finiteness growths of closed hyperbolic manifolds that totally geodesicall…
Average Distortion of Commensurators of Hyperbolic Groups
本文证明了几何刚性剩余有限双曲群的交换子具有有界平均畸变。
We examine residual properties of word-hyperbolic groups, adapting a method introduced by Darren Long to study the residual properties of Kleinian groups.
Coboundary expansion and Gromov hyperbolicity
本文证明了若紧致 $n$-流形允许一列在 $n-2$ 维形成余边界展开的剩余覆盖，则其具有 Gromov 双曲基本群，并指出非双曲紧致连通不可约 3-流形的剩余覆盖列不是 1-余边界展开子。
For a group G relatively hyperbolic to a family of residually finite groups satisfying the Farrell-Jones conjecture, we reduce the solution of the Farrell-Jones conjecture for G to the case of certain nice cyclic extensions in G.
Short hierarchically hyperbolic groups II: quotients and the Hopf property for Artin groups
本文证明大多数大类型和超几何类型的Artin群是Hopf群，并展示大部分五孔球面映射群的商是层次超几何的。
A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups
本研究提出基于双曲单纯复形作用的层级双曲群组合判据，证明映射类群模Dehn扭大幂次的商群为层级双曲群，构造非初等双曲商群并将相关有限商问题归约为特定双曲群的剩余有限性问题。
An automorphism $α$ of a group $G$ is normal if it fixes every normal subgroup of $G$ setwise. We give an algebraic description of normal automorphisms of relatively hyperbolic groups. In particular, we prove that for any relatively hyperbolic group $G$, $Inn(G)$ has finite index in the subgroup $Aut_n(G)$ of normal automorphisms. If, in addition, $G$ is non-elementary and has no non-trivial finite normal subgroups, then $Aut_n(G)=Inn(G)$. As an application, we show that $Out(G)$ is residually finite for every finitely generated residually finite group $G$ with more than one end.
We provide a solution to the isomorphism problem for torsion-free relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela); and (ii) fully residually free groups (Bumagin, Kharlampovich and Miasnikov). We also give a solution to the homeomorphism problem for finite volume hyperbolic n-manifolds, for $n \ge 3$. In the course of the proof of the main result, we prove that a particular JSJ decomposition of a freely indecomposable torsion-free relatively hyperbolic group with abelian parabo…
Every non-trivial knot group is fully residually perfect
本文证明每个非平凡纽结群均为完全剩余完美闭3-流形群，拓展了非平凡纽结群的完全剩余性质相关结论。
We make a few observations on the absence of geometric and topological rigidity for acylindrically hyperbolic and relatively hyperbolic groups. In particular, we demonstrate the lack of a well-defined limit set for acylindrical actions on hyperbolic spaces, even under the assumption of universality. We also prove a statement about relatively hyperbolic groups inspired by a remark by Groves, Manning, and Sisto about the quasi-isometry type of combinatorial cusps. Finally, we summarize these results in a table in order to assert a meta-statement about the decay of metric rigidity as the conditio…
We give the first examples of closed fibered hyperbolic 3-manifolds whose fundamental groups are distinguished from every other finitely generated, residually finite group by their finite quotients. One of the examples is also the first example of a non-orientable profinitely rigid hyperbolic 3-manifold.
Dehn fillings for relatively hyperbolic groups generalize the topological Dehn surgery on a non-compact hyperbolic $3$-manifold such as a hyperbolic knot complement. We prove a rigidity result saying that if two non-elementary relatively hyperbolic groups without suitable splittings have sufficiently many isomorphic Dehn fillings, then these groups are in fact isomorphic. Our main application is a solution to the isomorphism problem in the class of non-elementary relatively hyperbolic groups with residually finite parabolic groups and with no suitable splittings.
Connections between conjugation quandles and their underlying groups via residual finiteness and the Hopf property
本文研究了共轭quandle与其底层群的Hopf性质和剩余有限性联系，证明了Hopf性质的传递性并提供反例，同时利用Dehn quandle完全刻画了Baumslag-Solitar群共轭quandle的剩余有限性。
For n>3 we study spaces obtained from finite volume complete real hyperbolic n-manifolds by removing a compact totally geodesic submanifold of codimension two. We prove that their fundamental groups are relative hyperbolic, co-Hopf, biautomatic, residually hyperbolic, not Kähler, not isomorphic to lattices in virtually connected real Lie groups, have no nontrivial subgroups with property (T), have finite outer automorphism groups, satisfy Mostow-type Rigidity, have finite asymptotic dimension and rapid decay property, and satisfy Baum-Connes conjecture. We also characterize those lattices in r…
Let Gamma be a fixed hyperbolic group. The Gamma-limit groups of Sela are exactly the finitely generated, fully residually Gamma groups. We give a new invariant of Gamma-limit groups called Gamma-discriminating complexity and show that the Gamma-discriminating complexity of any Gamma-limit group is asymptotically dominated by a polynomial. Our proof relies on an embedding theorem of Kharlampovich-Myasnikov which states that a Gamma-limit group embeds in an iterated extension of centralizers over Gamma. The result then follows from our proof that if G is an iterated extension of centralizers ov…
Cohomology of group theoretic Dehn fillings II
本文研究群论Dehn填充的上同调，利用Cohen-Lyndon性质推导谱序列计算商群上同调，建立相对上同调同构并推广相关结果，还强化了柱形双曲群的SQ-泛性等结论。
Let $G$ be either a non-elementary (word) hyperbolic group or a large group (both in the sense of Gromov). In this paper we describe several approaches for constructing continuous families of periodic quotients of $G$ with various properties. The first three methods work for any non-elementary hyperbolic group, producing three different continua of periodic quotients of $G$. They are based on the results and techniques, that were developed by Ivanov and Olshanskii in order to show that there exists an integer $n$ such that $G/G^n$ is an infinite group of exponent $n$. The fourth approach start…
Solvability of the conjugacy problem for relatively hyperbolic groups was announced by Gromov [Hyperbolic groups, MSRI publications 8 (1987)]. Using the definition of Farb of a relatively hyperbolic group in the strong sense [B Farb, Relatively hyperbolic groups, Geom. Func. Anal. 8 (1998) 810-840], we prove this assertion. We conclude that the conjugacy problem is solvable for fundamental groups of complete, finite-volume, negatively curved manifolds, and for finitely generated fully residually free groups.
Deciding if a hyperbolic group splits over a given quasiconvex subgroup
本文提出判定双曲群沿给定拟凸子群是否分裂的算法，并扩展至计算过滤端数与Schreier图端数，利用带标号有向图编码边界连通性信息。
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 …
Actions of acylindrically hyperbolic groups on $\ell^1$
本文为具有双曲特征的群构造了 $\ell^1$ 和 $L^1$ 上的仿射一致 Lipschitz 作用，并证明对 acylindrically hyperbolic 群轨道无界，对剩余有限双曲群和映射类群轨道为真且诱导度量与字度量拟等距。
Let $Σ_{g,p}$ be the genus--$g$ oriented surface with $p$ punctures, with either $g>0$ or $p>3$. We show that $MCG(Σ_{g,p})/DT$ is acylindrically hyperbolic where $DT$ is the normal subgroup of the mapping class group $MCG(Σ_{g,p})$ generated by $K^{th}$ powers of Dehn twists about curves in $Σ_{g,p}$ for suitable $K$. Moreover, we show that in low complexity $MCG(Σ_{g,p})/DT$ is in fact hyperbolic. In particular, for $3g-3+p\leq 2$, we show that the mapping class group $MCG(Σ_{g,p})$ is fully residually non-elementary hyperbolic and admits an affine isometric action with unbounded orbits on s…
Let $H$ be a torsion-free $δ$-hyperbolic group with respect to a finite generating set $S$. Let $a_1,..., a_n$ and $a_{1*},..., a_{n*}$ be elements of $H$ such that $a_{i*}$ is conjugate to $a_i$ for each $i=1,..., n$. Then, there is a uniform conjugator if and only if $W(a_{1*},..., a_{n*})$ is conjugate to $W(a_1,..., a_n)$ for every word $W$ in $n$ variables and length up to a computable constant depending only on $δ$, $\sharp{S}$ and $\sum_{i=1}^n |a_i|$. As a corollary, we deduce that there exists a computable constant $\mathcal{C}=\mathcal{C}(δ, \sharp S)$ such that, for any endomorphism…
GAGTA 2023 Problem Session
本文整理了2023年GAGTA“群与动力学”会议问题研讨会上提出的8项群论相关研究问题，涵盖剩余有限性增长、双曲群模型等方向。
Stable cylinders and fine structures for hyperbolic groups and curve graphs
本文证明了剩余有限双曲群和有限型曲面曲线图具有全局稳定圆柱体，并通过推广Sageev构造提升精细性质，进而证明曲线图可等变拟等距嵌入有限拟树乘积。
Motivated by the results of Scott and Patel about "untangling" closed geodesics in finite covers of hyperbolic surfaces, we introduce and study primitivity, simplicity and non-filling index functions for finitely generated free groups. We obtain lower bounds for these functions and relate these free group results back to the setting of hyperbolic surfaces. An appendix by Khalid Bou-Rabee connects the primitivity index function to the residual finiteness growth function for $F_N$.
Drilling hyperbolic groups
本文通过钻孔超几何群沿特定子群，将Cannon猜想转化为相对版本，从而更易研究。
Suppose that all hyperbolic groups are residually finite. The following statements follow: In relatively hyperbolic groups with peripheral structures consisting of finitely generated nilpotent subgroups, quasiconvex subgroups are separable; Geometrically finite subgroups of non-uniform lattices in rank one symmetric spaces are separable; Kleinian groups are subgroup separable. We also show that LERF for finite volume hyperbolic 3-manifolds would follow from LERF for closed hyperbolic 3-manifolds. The method is to reduce, via combination and filling theorems, the separability of a quasiconvex s…
We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group $\mathrm{PSL}(2,\mathbb{Z}[ω])$ with $ω^2+ω+1=0$ is rigid in this sense. Other examples include the non-uniform lattice of minimal co-volume in $\mathrm{PSL}(2,\mathbb{C})$ and the fundamental group of the Weeks manifold (the closed hyperbolic $3$-manifold of minimal volume).
We prove that the outer automorphism group $Out(G)$ is residually finite when the group $G$ is virtually compact special (in the sense of Haglund and Wise) or when $G$ is isomorphic to the fundamental group of some compact $3$-manifold. To prove these results we characterize commensurating endomorphisms of acylindrically hyperbolic groups. An endomorphism $ϕ$ of a group $G$ is said to be commensurating, if for every $g \in G$ some non-zero power of $ϕ(g)$ is conjugate to a non-zero power of $g$. Given an acylindrically hyperbolic group $G$, we show that any commensurating endomorphism of $G$ i…
We study spaces obtained from a complete finite volume complex hyperbolic n-manifold M by removing a compact totally geodesic complex (n-1)-submanifold. The main result is that the fundamental group of M-S is relatively hyperbolic, relative to fundamental groups of the ends of M-S, and M-S admits a complete finite volume A-regular Riemannian metric of negative sectional curvature. It follows that for n>1 the fundamental group of M-S satisfies Mostow-type Rigidity, has finite asymptotic dimension and rapid decay property, satisfies Borel and Baum-Connes conjectures, is co-Hopf and residually hy…
Homology growth, hyperbolization, and fibering
本文提出保持双曲性与同调增长性质的双曲反射群技巧，并据此构造出不可实际上纤维化于圆周的 Gromov 双曲 7-流形。
Arithmetic trialitarian hyperbolic lattices are not LERF
本文证明了 $\mathbf{PSO}_{7,1}(\mathbb{R})$ 中的三元算术格不是 LERF，结合先前工作得出 $\mathbf{PO}_{n,1}(\mathbb{R})$（$n>3$）中所有算术格均非 LERF 的结论。
We prove that cubulated hyperbolic groups are virtually special. The proof relies on results of Haglund and Wise which also imply that they are linear groups, and quasi-convex subgroups are separable. A consequence is that closed hyperbolic 3-manifolds have finite-sheeted Haken covers, which resolves the virtual Haken question of Waldhausen and Thurston's virtual fibering question. An appendix to this paper by Agol, Groves, and Manning proves a generalization of the main result of "Residual finiteness, QCERF and fillings of hyperbolic groups".
Property QT of relatively hierarchically hyperbolic groups
本文通过建立充分条件，统一并推广了几类非正曲率群具有性质QT的相关结果。剩余有限且属于特定类群的群具有性质QT，还引入更强的性质QT\(_0\)并证明其在图乘积下不变。
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。
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。
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