Robert H. Gilman, S. Hermiller, Derek F. Holt, Sarah Rees
We prove that a finitely generated group $G$ is virtually free if and only if there exists a generating set for $G$ and $k > 0$ such that all $k$-locally geodesic words with respect to that generating set are geodesic.
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We give a new characterisation of virtually free groups using graph minors. Namely, we prove that a finitely generated, infinite group is virtually free if and only if for any finite generating set, the corresponding Cayley graph is minor excluded. This answers a question of Ostrovskii and Rosenthal. The proof relies on showing that a finitely generated group that is minor excluded with respect to every finite generating set is accessible, using a graph-theoretic characterisation of accessibility due to Thomassen and Woess.
This article is dedicated to the characterisation of the relative hyperbolicity of Haglund and Wise's special groups. More precise, we introduce a new combinatorial formalism to study (virtually) special groups, and we prove that, given a cocompact special group $G$ and a finite collection of subgroups $\mathcal{H}$, then $G$ is hyperbolic relative to $\mathcal{H}$ if and only if (i) each subgroup of $\mathcal{H}$ is convex-cocompact, (ii) $\mathcal{H}$ is an almost malnormal collection, and (iii) every non-virtually cyclic abelian subgroup of $G$ is contained in a conjugate of some group of $…
Virtually free-by-cyclic groups
本文在双曲且虚拟紧特殊的群中获得了虚拟自由-循环群的同调刻画，并以此证明所有带挠的单关系群均为虚拟自由-循环群，解决了Baumslag的猜想。
This note contains a (short) proof of the following generalisation of the Friedman--Mineyev theorem (earlier known as the Hanna Neumann conjecture): if $A$ and $B$ are nontrivial free subgroups of a virtually free group containing a free subgroup of index $n$, then $rank(A\cap B)-1\leqslant n\cdot(rank(A)-1)\cdot(rank(B)-1)$. In addition, we obtain a virtually-free-product analogue of this result.
In this note we prove that finitely generated virtually free groups are stable with respect to a normalized $p$-Schatten norm for $1\leq p < \infty$. In particular, this implies that virtually free groups are Hilbert-Schmidt stable.
In this short note we prove that a class of Artin groups of affine and complex types are virtually poly-free, answering partially the question if all Artin groups are virtually poly-free.
A finitely generated virtually free pro-p group with finite centralizers of its torsion elements is the free pro-p product of finite p-groups and a free pro-p factor.
An analogue of the strengthened Hanna Neumann conjecture for virtually free groups and virtually free products
本文将自由群中关于子群交的秩的强化汉纳-诺伊曼猜想（弗里德曼-米涅耶夫定理）推广到拟自由群及包含左序群自由积为有限指数子群的更一般群类。
A second countable virtually free pro-p group all of whose torsion elements have finite centralizer is the free pro-p product of finite p-groups and a free pro-p factor.
We prove that finitely generated virtually free groups are stable in permutations. As an application, we show that almost-periodic almost-automorphisms of labelled graphs are close to periodic automorphisms.
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.
On the virtually cyclic dimension of normally poly-free groups
本文给出了正规多自由群虚拟循环维数关于长度的上界，并由此证明FC型虚拟偶Artin群存在有限维分类空间模型。
$\mathbb{Z}^2$-free subgroups of Thompson's $T$ are virtually free
本文证明Thompson群T的有限生成且不含任意高阶阿贝尔群的子群为虚阿贝尔或虚自由，进而得出曲面群不能嵌入T，解答了Belk与Moore的公开问题。
We prove that every finitely generated, virtually solvable minimax group can be expressed as a homomorphic image of a virtually torsion-free, virtually solvable minimax group. This result enables us to generalize a theorem of Ch. Pittet and L. Saloff-Coste about random walks on finitely generated, virtually solvable minimax groups. Moreover, the paper identifies properties, such as the derived length and the nilpotency class of the Fitting subgroup, that are preserved in the covering process. Finally, we determine exactly which infinitely generated, virtually solvable minimax groups also posse…
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.
For a finitely generated group $Γ$ denote by $μ(Γ)$ the growth coefficient of $Γ$, that is, the infimum over all real numbers $d$ such that $s_n(Γ)<n!^d$. We show that the growth coefficient of a virtually free group is always rational, and that every rational number occurs as growth coefficient of some virtually free group. Moreover, we describe an algorithm to compute $μ$.
Any virtually free group $H$ containing no non-trivial finite normal subgroup (e.g., the infinite dihedral group) is a retract of any finitely generated group containing $H$ as a verbally closed subgroup.
We prove the pro-$p$ version of the Karras, Pietrowski, Solitar, Cohen and Scott result stating that a virtually free group acts on a tree with finite vertex stabilizers. If a virtually free pro-$p$ group $G$ has finite centralizes of all non-trivial torsion elements more stronger statement is proved: $G$ embeds into a free pro-$p$ product of a free pro-$p$ group and finite $p$-group. The integral $p$-adic representation theory is used in the proof; it replaces the Stallings theory of ends in the pro-$p$ case.
We prove an estimate for the rank of the intersection of free subgroups in virtually free groups, which is analogous to the Hanna Neumann inequality for subgroups in a free group and to the S.V. Ivanov estimate for subgroups in free products of groups. We also prove a more general estimate for the rank of the intersection of free subgroups in the fundamental group of a finite graph of groups with finite edge groups.
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.
We fill details in the proof of Lemma 13 in Bull.London.Math.Soc 40 (2008) (that is Lemma 3.2 in arXiv:0712.4244v1).
We define Euler characteristics on classes of residually finite and virtually torsion free groups and we show that they satisfy certain formulas in the case of amalgamated free products and HNN extensions over finite subgroups. These forumlas are obtained from a general result which applies to the rank gradient and the first $L^2$-Betti number of a finitely generated group.
The Stallings construction for finitely generated subgroups of free groups is generalized by introducing the concept of Stallings section, which allows an eficient computation of the core of a Schreier graph based on edge folding. It is proved that those groups admitting Stallings sections are precisely finitely generated virtually free groups, through a constructive approach based on Bass-Serre theory. Complexity issues and applications are also discussed.
It is shown that a finitely generated pro-p group G which is a virtually free pro-p product splits either as a free pro-p product with amalgamation or as a pro-p HNN-extension over a finite p-group. More precisely, G is the pro-p fundamental group of a finite graph of finitely generated pro-p groups with finite edge groups. This generalizes previous results of W. Herfort and the second author (cf. [2]).
A theorem of Myasnikov and Roman'kov says that any verbally closed subgroup of a finitely generated free group is a retract. We prove that all free (and many virtually free) verbally closed subgroups are retracts in any finitely generated group.
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.
We prove various results connecting structural or algebraic properties of graphs and groups to conditions on their spaces of harmonic functions. In particular: we show that a group with a finitely supported symmetric measure has a finite-dimensional space of harmonic functions if and only if it is virtually cyclic; we present a new proof of a result of V. Trofimov that an infinite vertex-transitive graph admits a non-constant harmonic function; we give a new proof of a result of T. Ceccherini-Silberstein, M. Coornaert and J. Dodziuk that the Laplacian on an infinite, connected, locally finite …
A group is tubular if it acts on a tree with $\mathbb{Z}^2$ vertex stabilizers and $\mathbb{Z}$ edge stabilizers. We prove that a tubular group is virtually special if and only if it acts freely on a locally finite CAT(0) cube complex. Furthermore, we prove that if a tubular group acts freely on a finite dimensional CAT(0) cube complex, then it virtually acts freely on a three dimensional CAT(0) cube complex.
In this paper we consider some results obtained for graphs using minimal vertex separators and generalized chordality and translate them to the context of Geometric Group Theory. Using these new tools, we are able to give two new characterizations for a group to be virtually free. Furthermore, we prove that the Baumslag-Solitar group $BS(1,n)$ is $k$-chordal for some $k$ if and only if $|n|<3$ and we give an application of generalized chordality to the study of the word problem.
The article deals with profinite groups in which centralizers are virtually procyclic. Suppose that G is a profinite group such that the centralizer of every nontrivial element is virtually torsion-free while the centralizer of every element of infinite order is virtually procyclic. We show that G is either virtually pro-p for some prime p or virtually torsion-free procyclic. The same conclusion holds for profinite groups in which the centralizer of every nontrivial element is virtually procyclic; moreover, if G is not pro-p, then G has finite rank.
We consider the subgroup of points of finite orbit through the action of an endomorphism of a virtually free group, with particular emphasis on the subgroup of eventually fixed points, EvFix($φ$): points whose orbit contains a fixed point. We provide an algorithm to compute the subgroup of fixed points of an endomorphism of a finitely generated virtually free group and prove that finite orbits have cardinality bounded by a computable constant, which allows us to solve several algorithmic problems: deciding if $φ$ is a finite order element of End($G$), if $φ$ is aperiodic, if EvFix($φ$) is fini…
Property (LR) and an embedding theorem for virtually free groups
本文证明了每个虚自由群G具有Long和Reid提出的性质(LR)，即每个有限生成子群都是有限指数子群的 retract。主要方法是证明每个可数虚自由群可嵌入到有限群的双群中。
Let $Γ$ be a group which is virtually free of rank at least 2 and let $\mathcal{F}_{td}(Γ)$ be the family of totally disconnected, locally compact groups containing $Γ$ as a co-compact lattice. We prove that the values of the scale function with respect to groups in $\mathcal{F}_{td}(Γ)$ evaluated on the subset $Γ$ have only finitely many prime divisors. This can be thought of as a uniform property of the family $\mathcal{F}_{td}(Γ)$.
A group G is called subgroup conjugacy separable (abbreviated as SCS), if any two finitely generated and non-conjugate subgroups of G remain non-conjugate in some finite quotient of G. We prove that the free groups and the fundamental groups of finite trees of finite groups with some normalizer condition are SCS. We also introduce the subgroup into-conjugacy separability property and prove that the above groups have this property too.
A fast algorithm for Stallings foldings over virtually free groups
本研究提出一种求解虚自由群子群成员问题的快速算法，时间复杂度接近线性，可用于矩阵群计算，还能判定有限生成子群是否同构于自由群。
We prove that two co-Hopfian finitely generated virtually free groups are elementarily equivalent if and only if they are isomorphic. We also prove that co-Hopfian finitely generated virtually free groups are homogeneous in the sense of model theory.
Division rings for group algebras of virtually compact special groups and $3$-manifold groups
本文为无挠虚拟紧特殊群和有限生成无挠3维流形群的群代数构造除环嵌入，证明其在局部可指示时为Hughes自由，并推出零因子猜想、Kielak–Linton猜想及凝聚性等结论。
We give a complete classification of finitely generated virtually free groups up to $\forall\exists$-elementary equivalence. As a corollary, we give an algorithm that takes as input two finite presentations of virtually free groups, and decides whether these groups have the same $\forall\exists$-theory or not.
We give an elementary criterion on a group G for the map from Aut(G) to Out(G) to split virtually. This criterion applies to many residually finite CAT(0) groups and hyperbolic groups, and in particular to all finitely generated Coxeter groups. As a consequence the outer automorphism group of any finitely generated Coxeter group is residually finite and virtually torsion-free.
Non-solvable torsion-free virtually solvable groups
本文研究非可解、无挠、几乎可解群的 Hirsch 长度下界，证明其至少为 10，并分析了不同长度范围内可能出现的单因子类型。
Combinatorial Characterizations of Virtually Torsion-Free and Virtually Free Groups
本文通过图分解理论，研究了虚 torsion 自由和虚自由群的组合特征，并建立了相应的等价条件。
We construct cocompact lattices in a product of trees which are not virtually torsion-free. This gives the first examples of hierarchically hyperbolic groups which are not virtually torsion-free
We determine the asymptotic number of index $n$ subgroups in virtually cyclic right-angled Coxeter groups and their free products as $n\to\infty$.
We study the problem of determining the isomorphism classes of the virtually cyclic subgroups of the n-string braid groups B_n(S^2) of the 2-sphere S^2. If n is odd, or if n is even and sufficiently large, we obtain the complete classification. For small even values of n, the classification is complete up to an explicit finite number of open cases. In order to prove our main theorem, we obtain a number of other results of independent interest, notably the characterisation of the centralisers and normalisers of the finite cyclic and dicyclic subgroups of B_n(S^2), a result concerning conjugate …
For every free product decomposition $G = G_{1} \ast ...\ast G_{q} \ast F_{r}$, where $F_r$ is a finitely generated free group, of a group $G$ of finite Kurosh rank, we can associate some (relative) outer space $\mathcal{O}$. In this paper, we develop the theory of (stable) laminations for (relative) irreducible with irreducible powers (IWIP) automorphisms. In particular, we examine the action of $Out(G, \mathcal{O}) \leq Out(G)$ (i.e. the automorphisms which preserve the set of conjugacy classes of $G_i$'s) on the set of laminations. We generalise the theory of the attractive laminations asso…
Let $Q_m$ be the HNN extension of $\Z/m \times \Z/m$ where the stable letter conjugates the first factor to the second. We explore small presentations of the groups $Γ_{m,n}=Q_m \ast Q_n$. We show that for certain choices of (m,n), for example (2,3), the group $Γ_{m,n}$ has a relation gap unless it admits a presentation with at most 3 defining relations, and we establish restrictions on the possible form of such a presentation. We then associate to each (m,n) a 3-complex with 16 cells. This 3-complex is a counterexample to the D(2) conjecture if $Γ_{m,n}$ has a relation gap.
A non-plain virtually free group with a strictly 2-geodetic Cayley graph
本文构造了一个无限几乎自由群$G=C_4\ast_{C_2}D_3$，其Cayley图是2-测地但非测地，反驳了相关猜想，并证明其自由不可分解性。
We give a description of elementary subgroups (in the sense of first-order logic) of finitely generated virtually free groups. In particular, we recover the fact that elementary subgroups of finitely generated free groups are free factors. Moreover, we give an algorithm that takes as input a finite presentation of a virtually free group $G$ and a finite subset $X$ of $G$, and decides if the subgroup of $G$ generated by $X$ is $\exists\forall\exists$-elementary. We also prove that every elementary embedding of an equationally noetherian group into itself is an automorphism.
We show that if a group is not virtually cyclic and is hyperbolic relative to a family of proper subgroups, then it has a hyperbolically embedded subgroup which contains a finitely generated non-abelian free group as a finite index subgroup.
Cleanliness versus Specialness
该研究通过证明有限秩自由群的几何洁净图基本群不必为虚紧特殊群，回答了Wise的问题，明确了虚VH-洁净图类是虚几何洁净图类的真子类，推进了群论相关分类研究。
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.
Cops and robbers for hyperbolic and virtually free groups
本文定义强、弱警察数两种群不变量，证明双曲群对应强警察数为1、虚拟自由群对应弱警察数为1，还给出点灯人群等的强警察数计算实例，解答了相关公开问题。
We give a geometric proof of a well known theorem that describes splittings of a free group as an amalgamated product or HNN extension over the integers. The argument generalizes to give a similar description of splittings of a virtually free group over a virtually cyclic group.
Geodesic languages for rational subsets and conjugates in virtually free groups
本文证明拟自由群中有理子集与测地词语言的有理性等价，并由此得出有理子集的双重广义共轭问题可判定，自由群情形下对共轭元施加有理约束仍可判定。
In this short note, we show that a group acting geometrically on a CAT(0) cube complex with virtually abelian hyperplane-stabilisers must decompose virtually as a free product of free abelian groups and surface groups.
Balanced groups and the virtually cyclic dimension of poly-surfaces groups
本文通过研究平衡性质（Lück的条件C），获得正常多面体群和正常多自由群的虚拟循环维数的线性上界，证明了平衡性质在短正合列、极限和圆柱形图群分解下的稳定性，并应用该结果证明了正常多超几何群、正常多自由群和正常多面体群均为平衡群。
Free groups are known to be homogeneous, meaning that finite tuples of elements which satisfy the same first-order properties are in the same orbit under the action of the automorphism group. We show that virtually free groups have a slightly weaker property, which we call uniform almost-homogeneity: the set of $k$-tuples which satisfy the same first-order properties as a given $k$-tuple $\mathbf{u}$ is the union of a finite number of $\mathrm{Aut}(G)$-orbits, and this number is bounded independently from $\mathbf{u}$ and $k$. Moreover, we prove that there exists a virtually free group which i…
Using a probabilistic argument we show that the second bounded cohomology of an acylindrically hyperbolic group $G$ (e.g., a non-elementary hyperbolic or relatively hyperbolic group, non-exceptional mapping class group, ${\rm Out}(F_n)$, \dots) embeds via the natural restriction maps into the inverse limit of the second bounded cohomologies of its virtually free subgroups, and in fact even into the inverse limit of the second bounded cohomologies of its hyperbolically embedded virtually free subgroups. This result is new and non-trivial even in the case where $G$ is a (non-free) hyperbolic gro…
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。
On generalized (twisted) conjugacy separability of some extensions of groups
引入广义共轭可分性性质，证明群$G$在有限与循环扩张下扭曲共轭、Brinkmann共轭等可分性的等价与蕴含关系，并应用于自由群等群类。
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