Will Dison, Murray Elder, Timothy R Riley
We prove that $n^{7/3}$ is an isoperimetric function for a group of Stallings that is finitely presented but not of type $\mathcal{F}_3$. Note: The authors with Robert Young have now proved a quadratic Dehn function for this group. See arXiv:0712.3877
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Isoperimetric behavior of generalized Stallings-Bieri groups
本文引入群到有限秩自由阿贝尔群满射的 $n$-split 概念，为有限展示群直积的某些共阿贝尔子群的 Dehn 函数提供界，这些子群包含并显著推广了 Stallings-Bieri 群。
We show that the Stallings-Bieri groups, along with certain other Bestvina-Brady groups, have quadratic Dehn function.
We prove that the Dehn function of a group of Stallings that is finitely presented but not of type F_3 is quadratic. To appear in Geometric and Functional Analysis.
We re-cast in a more combinatorial and computational form the foldings approach of John Stallings and pursue a detailed study of the subgroup structure of free groups. In particular, we introduce the notions of an "algebraic" and a "free" extension for subgroups of free groups and apply these notions to obtain a number of algebraic and algorithmic results.
The celebrated Stallings' decomposition theorem states that the splitting of a finite index subgroup $H$ of a finitely generated group $G$ as an amalgamated free product or an HNN-extension over a finite group implies the same for $G$. We generalize the pro-$p$ version of it proved by Weigel and the second author to splittings over infinite pro-$p$ groups. This generalization does not have any abstract analogs. We also prove that generalized accessibility of finitely generated pro-$p$ groups is closed for commensurability.
This paper studies the free group of rank two from the point of view of Stallings core graphs. The first half of the paper examines primitive elements in this group, giving new and self-contained proofs for various known results about them. In particular, this includes the classification of bases of this group. The second half of the paper is devoted to constructing a counterexample to a conjecture by Miasnikov, Ventura and Weil, which seeks to characterize algebraic extensions in free groups in terms of Stallings graphs.
We show that one can define and effectively compute Stallings graphs for quasi-convex subgroups of automatic groups (\textit{e.g.} hyperbolic groups or right-angled Artin groups). These Stallings graphs are finite labeled graphs, which are canonically associated with the corresponding subgroups. We show that this notion of Stallings graphs allows a unified approach to many algorithmic problems: some which had already been solved like the generalized membership problem or the computation of a quasi-convexity constant (Kapovich, 1996); and others such as the computation of intersections, the con…
We describe a higher dimensional analogue of the Stallings folding sequence for group actions on CAT(0) cube complexes. We use it to give a characterization of quasiconvex subgroups of hyperbolic groups which act properly co-compactly on CAT(0) cube complexes via finiteness properties of their hyperplane stabilizers.
We extend the classical Stallings theory (describing subgroups of free groups as automata) to direct products of free and abelian groups: after introducing enriched automata (i.e., automata with extra abelian labels), we obtain an explicit bijection between subgroups and a certain type of such enriched automata, which - as it happens in the free group - is computable in the finitely generated case. This approach provides a neat geometric description of (even non finitely generated) intersections of finitely generated subgroups within this non-Howson family. In particular, we give a geometric s…
In the 1980's Stallings showed that every finitely generated subgroup of a free group is canonically represented by a finite minimal immersion of a bouquet of circles. In terms of the theory of automata, this is a minimal finite inverse automaton. This allows for the deep algorithmic theory of finite automata and finite inverse monoids to be used to answer questions about finitely generated subgroups of free groups. In this paper we attempt to apply the same methods to other classes of groups. A fundamental new problem is that the Stallings folding algorithm must be modified to allow for ``sew…
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.
Geometric methods proposed by Stallings for treating finitely generated subgroups of free groups were successfully used to solve a wide collection of decision problems for free groups and their subgroups. It turns out that Stallings' methods can be effectively generalized for the class of amalgams of finite groups. In the present paper we employ subgroup graphs constructed by the generalized Stallings' folding algorithm to solve various algorithmic problems in amalgams of finite groups.
Colored Stallings graphs and Counterexamples to Stallings equalizer conjecture
本文通过引入着色Stallings图，证明对于n≥2，存在两个单射g,h:F_n→F_2，使得等式集的秩≥2n-2，从而推翻Stallings等式猜想。
We show that for a fixed free group F and an arbitrary finitely generated subgroup H (as given above) we can perform the Stalling's folding process in time O(N log^*(N)), where N is the sum of the word lengths of the given generators of H.
In this paper we review some of the fundamental properties of the free group and give a detailed account of Stallings's theory of automata, a geometric interpretation of its subgroups that has been (and still is) immensely fruitful, both as a means of understanding classical results, as as a source of new results. We review some of the most important ones. -- - En aquest article revisem algunes de les propietats fonamentals del group lliure i fem una exposició detallada de la teoria dels autòmats de Stallings, una interpretació geomètrica dels seus subgrups que ha estat (i segueix essent) imme…
Stallings' Group is Simply Connected at Infinity
本文验证了关于Bieri-Stallings群在无穷远处连通性的猜想，证明了Stallings群$B_3$在无穷远处是单连通的。
This survey is intended to be a fast (and reasonably updated) reference for the theory of Stallings automata and its applications to the study of subgroups of the free group, with the main accent on algorithmic aspects. Consequently, results concerning finitely generated subgroups have greater prominence in the paper. However, when possible, we try to state the results with more generality, including the usually overlooked non-(finitely-generated) case.
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。
Geometric methods proposed by Stallings for treating finitely generated subgroups of free groups were successfully used to solve a wide collection of decision problems for free groups and their subgroups. In the present paper we employ the generalized Stallings' methods, developed by the author, to solve various algorithmic problems concerning finitely generated subgroups of amalgams of finite groups.
Birget, Margolis, Meakin and Weil proved that a finitely generated subgroup $K$ of a free group is pure if and only if the transition monoid $M(K)$ of its Stallings automaton is aperiodic. In this paper, we establish further connections between algebraic properties of $K$ and algebraic properties of $M(K)$. We mainly focus on the cases where $M(K)$ belongs to the pseudovariety $\overline{\boldsymbol{\mathbf{H}}}$ of finite monoids all of whose subgroups lie in a given pseudovariety $\overline{\boldsymbol{\mathbf{H}}}$ of finite groups. We also discuss normal, malnormal and cyclonormal subgroup…
Quadratic Isoperimetric Inequality for Mapping Tori of Hyperbolic Groups
该研究确立了一端无挠双曲群的映射环面具有二次等周不等式，推进了几何群论相关问题的研究。
We discuss properties of the complete Euler characteristic of a group G of type FP over the complex numbers and we relate it to the L2-Euler characteristic of the centralizers of the elements of G.
If $F$ is a finitely generated free group and $ϕ$ is a positive automorphism of $F$ then $F\rtimes_ϕZ$ satisfies a quadratic isoperimetric inequality.
We show that arithmetic subgroups of semisimple groups of relative Q-type A_n, B_n, C_n, D_n, E_6, or E_7 have an exponential lower bound to their isoperimetric inequality in the dimension that is 1 less than the real rank of the semisimple group.
We prove that every finitely generated nilpotent group of class c admits a polynomial isoperimetric function of degree c+1 and a linear upper bound on its filling length function.
On the Isoperimetric functions of a class of Artin Groups
本文针对定义图中不含标号为3的边的Artin群，利用相对扩展表示的small cancellation理论，构造出n^6多项式等周函数，从而证明偶Artin群的字问题可解。
Stallings foldings for rational subsets of automatic groups
本研究针对自动群的有理子集，在Kharlampovich等关于子群的工作基础上，提出基于Stallings折叠构造识别自动机的方法，针对子幺半群可检验构造完备性，结合小消去理论应用于亏格大于1的曲面群。
If F is a finitely generated free group and ϕis an automorphism of F then the mapping torus of ϕadmits a quadratic isoperimetric inequality. This is the third and final paper in a series proving this theorem. The first two were math.GR/0211459 and math.GR/0507589.
We prove that the word problem of a finitely generated group $G$ is in NP (solvable in polynomial time by a non-deterministic Turing machine) if and only if this group is a subgroup of a finitely presented group $H$ with polynomial isoperimetric function. The embedding can be chosen in such a way that $G$ has bounded distortion in $H$.
We provide a new method of constructing non-quasiconvex subgroups of hyperbolic groups by utilizing techniques inspired by Stallings' foldings. The hyperbolic groups constructed are in the natural class of right-angled Coxeter groups (RACGs for short) and can be chosen to be 2-dimensional. More specifically, given a non-quasiconvex subgroup of a (possibly non-hyperbolic) RACG, our construction gives a corresponding non-quasiconvex subgroup of a hyperbolic RACG. We use this to construct explicit examples of non-quasiconvex subgroups of hyperbolic RACGs including subgroups whose generators are a…
The isoperimeric spectrum consists of all real positive numbers $α$ such that $O(n^α)$ is the Dehn function of a finitely presented group. In this note we show how a recent result of Olshanskii completes the description of the isoperimetric spectrum modulo the celebrated Computer Science conjecture (and one of the seven Millennium Problems) $\mathbf{P=NP}$ and even a formally weaker conjecture.
Geometric methods proposed by Stallings for treating finitely generated subgroups of free groups were successfully used by many authors to solve a wide collection of decision problems for free groups and their subgroups. In the present paper we employ our generalized Stallings' folding method to introduce a procedure, which given a subgroup H of a free product of finite groups reads off its Kurosh decomposition from the subgroup graph of H.
If F is a finitely generated free group and ϕis an automorphism of F then F \rtimes_ϕZ satisfties a quadratic isoperimetric inequality.
We describe a general technique for embedding certain amalgamated products into direct products. This technique provides us with a way of constructing a host of finitely presented subgroups of automatic groups which are not even asynchronously automatic. We can also arrange that such subgroups satisfy, at best, an exponential isoperimetric inequality.
We show that there is no uniform polynomial isoperimetric function for finitely presented subgroups of direct products of free groups, by producing a sequence of subgroups $G_r\leq F_2^{(1)} \times \dots \times F_2^{(r)}$ of direct products of 2-generated free groups with Dehn functions bounded below by $n^{r}$. The groups $G_r$ are obtained from the examples of non-coabelian subdirect products of free groups constructed by Bridson, Howie, Miller and Short. As a consequence we obtain that residually free groups do not admit a uniform polynomial isoperimetric function.
We show that the generation problem in Thompson group $F$ is decidable, i.e., there is an algorithm which decides if a finite set of elements of $F$ generates the whole $F$. The algorithm makes use of the Stallings $2$-core of subgroups of $F$, which can be defined in an analogue way to the Stallings core of subgroups of a finitely generated free group. Further study of the Stallings $2$-core of subgroups of $F$ provides a solution to another algorithmic problem in $F$. Namely, given a finitely generated subgroup $H$ of $F$, it is decidable if $H$ acts transitively on the set of finite dyadic …
Let $Γ$ be a finite graph, and for each vertex $i$ let $G_i$ be a finitely presented group. Let $G$ be the graph product of the $G_i$. That is, $G$ is the group obtained from the free product of the $G_i$ by factoring out by the smallest normal subgroup containing all $[g,h]$ where $g\in G_i$ and $h\in G_j$ and there is an edge joining i and j . We show that $G$ has an isoperimetric function of degree $k\ge 2$ (or an exponential isoperimetric function) if each vertex group has such an isoperimetric function.
In this thesis we investigate the Dehn functions of two different classes of groups: subdirect products, in particular subdirect products of limit groups; and Bestvina-Brady groups. Let D = Γ_1 \times ... \times Γ_n be a direct product of n \geq 3 finitely presented groups and let H be a subgroup of D. Suppose that each Γ_i contains a finite index subgroup Γ_i' \leq Γ_i such that the commutator subgroup [D', D'] of D' = Γ_1' \times ... \times Γ_n' is contained in H. Suppose furthermore that, for each i, the subgroup Γ_i H has finite index in D. We prove that H is finitely presented and satisfi…
The idea of applying isoperimetric functions to group theory is due to M.Gromov. We introduce the concept of a ``bicombing of narrow shape'' which generalizes the usual notion of bicombing. Our bicombing is related to but different from the combings defined by M. Bridson. If the Cayley graph of a group with respect to a given set of generators admits a bicombing of narrow shape then the group is finitely presented and satisfies a sub-exponential isoperimetric inequality, as well as a polynomial isodiametric inequality. We give an infinite class of examples which are not bicombable in the usual…
We associate cube complexes called completions to each subgroup of a right-angled Coxeter group (RACG). A completion characterizes many properties of the subgroup such as whether it is quasiconvex, normal, finite-index or torsion-free. We use completions to show that reflection subgroups are quasiconvex, as are one-ended Coxeter subgroups of a 2-dimensional RACG. We provide an algorithm that determines whether a given one-ended, 2-dimensional RACG is isomorphic to some finite-index subgroup of another given RACG. In addition, we answer several algorithmic questions regarding quasiconvex subgro…
We provide the details for Gromov's proof of Stallings' theorem on groups with infinitely many ends using harmonic functions. The main technical result of the paper is a compactness theorem for a certain family of harmonic functions.
It is shown that there exist infinitely many non-integers $r>2$ such that the Dehn function of some finitely presented group is $\simeq n^r$. For each positive rational number $s$ we construct pairs of finitely presented groups $H\subset G$ such that the distortion of $H$ in $G$ is $\simeq n^s$. And for each $s\ge 1$ we also construct finitely presented groups whose isodiametric function is $\simeq n^s$.
Isoperimetric inequalities in finitely generated groups
本文研究了有限生成群的Dehn谱，计算了多种群的Dehn谱，并证明了存在不可数多个非准等距的有限生成群。
For every pair of positive integers $p > q$ we construct a one-relator group $R_{p,q}$ whose Dehn function is $\simeq n^{2 α}$ where $α= \log_2(2p / q)$. The group $R_{p,q}$ has no subgroup isomorphic to a Baumslag-Solitar group $BS(m,n)$ with $m \neq \pm n$, but is not automatic, not CAT(0), and cannot act freely on a CAT(0) cube complex. This answers a long-standing question on the automaticity of one-relator groups and gives counterexamples to a conjecture of Wise.
Towards a Stallings-type theorem for finite groups
该研究借助图子式理论中的局部分隔子与局部覆盖，发现并证明了有限幂零群的低阶Stallings型定理，明确了群的Cayley图局部覆盖端数、局部分隔子与群结构的等价关系。
In a seminal paper, Stallings introduced folding of morphisms of graphs. One consequence of folding is the representation of finitely-generated subgroups of a finite-rank free group as immersions of finite graphs. Stallings's methods allow one to construct this representation algorithmically, giving effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani--Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on CAT(0) cube complexes. In this paper we extend their techniques to fund…
On Finiteness of Homological Isoperimetric Functions on Top Dimensions
本文研究了在上维空间中群的同调等周函数有限性，证明了该函数要么线性要么无限，并指出其与超几何性相关。
This is a new and short proof of the main theorem of classical structure tree theory. Namely, we show the existence of certain automorphism-invariant tree-decompositions of graphs based on the principle of removing finitely many edges. This was first done in "Cutting up graphs" by M.J. Dunwoody. The main ideas are based on the paper "Vertex cuts" by M.J. Dunwoody and the author. We extend the theorem to a detailed combinatorial proof of J.R. Stallings' theorem on the structure of finitely generated groups with more than one end.
We show that word-hyperbolic groups satisfy linear isoperimetric functions for all homotopy types of surface diagrams. This generalises the linear isoperimetric functions for disc and annular diagrams.
We study the group of ends of a pro-p group G and prove a pro-p analog of Stallings' decomposition theorem.
Stallings's Fibring Theorem and $\mathrm{PD}^3$-pairs
本研究给出相对自足的证明，验证代数纤维化且属于$\mathrm{PD}^3$-对的群是紧非球面纤维化3-流形的基本群，进而为Stallings经典定理提供同调证明。
The goal of this note is to generalize Isoperimetric Inequality for random groups to the class of non-planar diagrams of bounded number of faces.
``What aspects of a group are unchanged, or stable, under homology equivalences''? The model theorem in this regard is the 1963 result of J. Stallings that the lower central series is preserved under any integral homological equivalence of groups. Various other theorems of this nature have since appeared. Stallings himself proved similar theorems for homology with rational or mod p coefficients. These involved different series of groups- variations of the lower central series. W. Dwyer generalized Stallings' integral results to larger classes of maps, work that was completed in the other cases…
We present a sharp version of the isoperimetric inequality for finitely generated groups due to T. Couhlon and L. Saloff-Coste based on the proof suggested by M. Gromov.
Ideals of equations for elements in a free group and Stallings folding
本文研究自由群中元素在子群系数下的方程理想，给出基于Stallings折叠的生成元算法，并刻画最小次数方程及次数集合，回答Rosenmann-Ventura问题。
We show that the mapping class group of a handlebody of genus at least 2 has a Dehn function of at most exponential growth type.
We introduce a combinatorial version of Stallings-Bestvina-Feighn-Dunwoody folding sequences. We then show how they are useful in analyzing the solvability of the uniform subgroup membership problem for fundamental groups of graphs of groups. Applications include coherent right-angled Artin groups and coherent solvable groups.
We study a large class of amenable locally compact groups containing all solvable algebraic groups over a local field and their discrete subgroups. We show that the isoperimetric profile of these groups is in some sense optimal among amenable groups. We use this fact to compute the probability of return of symmetric random walks, and to derive various other geometric properties which are likely to be only satisfied by these groups.
Unimodular totally disconnected locally compact groups of rational discrete cohomological dimension one
本文将 Stallings--Swan 定理推广至全不连通局部紧群，证明有理离散上同调维数不超过1的紧生成群必同构于有限profinite群图的基本群。
The Hattori-Stallings rank, the Euler-Poincaré characteristic and zeta functions of totally disconnected locally compact groups
该研究针对幺模全不连通局部紧群引入Hattori-Stallings秩类似物，定义有理离散Euler-Poincaré示性数，还构造了相关形式Dirichlet级数并建立其与示性数的恒等关系。
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