Groups with quadratic-non-quadratic Dehn functions

Algorithmic problems in groups with quadratic Dehn function

We construct and study finitely presented groups with quadratic Dehn function (QD-groups) and present the following applications of the method developed in our recent papers. (1) The isomorphism problem is undecidable in the class of QD-groups. (2) For every recursive function $f$, there is a QD-group $G$ containing a finitely presented subgroup $H$ whose Dehn function grows faster than $f$. (3) There exists a group with undecidable conjugacy problem but decidable power conjugacy problem; this group is QD.

自由群直积子群的 Dehn 函数。第二部分：精确计算

Dehn functions of subgroups of products of free groups. Part II: Precise computations

本文精确计算了 Bridson-Dison 群的 Dehn 函数为四次，并证明自由群直积中满足特定有限性型和余秩条件的余阿贝尔子群具有二次 Dehn 函数。

On the Dehn functions of Kähler groups

We address the problem of which functions can arise as Dehn functions of Kähler groups. We explain why there are examples of Kähler groups with linear, quadratic, and exponential Dehn function. We then proceed to show that there is an example of a Kähler group which has Dehn function bounded below by a cubic function and above by $n^6$. As a consequence we obtain that for a compact Kähler manifold having non-positive holomorphic bisectional curvature does not imply having quadratic Dehn function.

Partial Dehn twists of free groups relative to local Dehn twists - a dichotomy

A criterion for quadratic or higher growth of group automorphisms is established which are represented by graph-of-groups automorphisms with certain well specified properties. As a consequence, it is derived (using results of a previous paper of the author) that every partial Dehn twist automorphism of $\FN$ relative to local Dehn twist automorphisms is either an honest Dehn twist automorphism, or else has quadratic growth.

Homological Dehn functions of groups of type $FP_2$

We prove foundational results for homological Dehn functions of groups of type $FP_2$ such as superadditivity and the invariance under quasi-isometry. We then study the homological Dehn functions of Leary's groups $G_L(S)$ providing methods to obtain uncountably many groups with a given homological Dehn function. This allows us to show that there exist groups of type $FP_2$ with quartic homological Dehn function and unsolvable word problem.

Recognizing a relatively hyperbolic group by its Dehn fillings

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.

Subgroups of the mapping class group of the torus generated by powers of Dehn twists

We study subgroups of the mapping class group of the torus generated by powers generated by powers of Dehn twists. We give a criterion to show when a collection of powers Dehn twists generates a free group using the ping pong lemma. We show that the subgroup generated by three uniform powers of Dehn twists can be either the whole mapping class group, a direct product of a free group of rank two with the cyclic group of order two, or a free group of rank at most three. We characterize subgroups generated by a collection of (respectively squares of) Dehn twists, if there is a pair among so that …

The Dehn function of SL(n;Z)

We prove that when n >= 5, the Dehn function of SL(n;Z) is quadratic. The proof involves decomposing a disc in SL(n;R)/SO(n) into triangles of varying sizes. By mapping these triangles into SL(n;Z) and replacing large elementary matrices by "shortcuts," we obtain words of a particular form, and we use combinatorial techniques to fill these loops.

群与可定向曲面的Dehn quandle

Dehn quandles of groups and orientable surfaces

本文引入群关于子集的Dehn quandle以统一多种quandle构造，证明其包络群性质，研究可序性，并计算可定向曲面Dehn quandle的生成集与自同构群，推广了若干已知结果。

Dehn functions of higher rank arithmetic groups of type $A_n$ in products of simple Lie groups

Suppose $Γ$ is an arithmetic group defined over a global field $K$, that the $K$-type of $Γ$ is $A_n$ with $n \geq 2$, and that the ambient semisimple group that contains $Γ$ as a lattice has at least two noncocompact factors. We use results from Bestvina-Eskin-Wortman and Cornulier-Tessera to show that $Γ$ has a polynomially bounded Dehn function.

Scaled relators and Dehn functions for nilpotent groups

A homogeneous nilpotent Lie group has a scaling automorphism determined by a grading of its Lie algebra. Many proofs of upper bounds for the Dehn function of such a group depend on being able to fill curves with discs compatible with this grading; the area of such discs changes predictably under the scaling automorphism. In this paper, we present combinatorial methods for finding such bounds. Using this method, we give new proofs of some results on Dehn functions of nilpotent groups, prove theorems on central powers and certain quotients of nilpotent groups, and construct the first example of …

A Subgroup of a Direct Product of Free Groups whose Dehn Function has a Cubic Lower Bound

We establish a cubic lower bound on the Dehn function of a certain finitely presented subgroup of a direct product of 3 free groups.

The Geometry of the Handlebody Groups II: Dehn functions

We show that the Dehn function of the handlebody group is exponential in any genus $g\geq 3$. On the other hand, we show that the handlebody group of genus $2$ is cubical, biautomatic, and therefore has a quadratic Dehn function.

Superexponential Dehn functions inside CAT(0) groups

We construct 4-dimensional CAT(0) groups containing finitely presented subgroups whose Dehn functions are $\exp^{(n)}(x^m)$ for integers $n, m \geq 1$ and 6-dimensional CAT(0) groups containing finitely presented subgroups whose Dehn functions are $\exp^{(n)}(x^α)$ for integers $n \geq 1$ and $α$ dense in $[1,\infty)$. This significantly expands the known geometric behavior of subgroups of CAT(0) groups.

Dehn functions and Hölder extensions in asymptotic cones

The Dehn function measures the area of minimal discs that fill closed curves in a space; it is an important invariant in analysis, geometry, and geometric group theory. There are several equivalent ways to define the Dehn function, varying according to the type of disc used. In this paper, we introduce a new definition of the Dehn function and use it to prove several theorems. First, we generalize the quasi-isometry invariance of the Dehn function to a broad class of spaces. Second, we prove Hölder extension properties for spaces with quadratic Dehn function and their asymptotic cones. Finally…

Fully irreducible Automorphisms of the Free Group via Dehn twisting in $\sharp_k(S^2 \times S^1)$

By using a notion of a geometric Dehn twist in $\sharp_k(S^2 \times S^1)$, we prove that when projections of two $\mathbb{Z}$-splittings to the free factor complex are far enough from each other in the free factor complex, Dehn twist automorphisms corresponding to the $\mathbb{Z}$-splittings generate a free group of rank $2$. Moreover, every element from this free group is either conjugate to a power of one of the Dehn twists or it is a fully irreducible outer automorphism of the free group. We also prove that, when projected to the intersection graph, the same group of Dehn twists produce ato…

A small generating set for the twist subgroup of the mapping class group of a non-orientable surface by Dehn twists

We give a small generating set for the twist subgroup of the mapping class group of a non-orientable surface by Dehn twists. The difference between the number of the generators and a lower bound of numbers of generators for the twist subgroup by Dehn twists is one. The lower bounds is obtained from an argument of Hirose [5].

Dehn functions and finiteness properties of subgroups of perturbed right-angled Artin groups

We introduce the class of perturbed right-angled Artin groups. These are constructed by gluing Bieri double groups into standard right-angled Artin groups. As a first application of this construction we obtain families of CAT(0) groups containing finitely presented subgroups which are not of type $\mathrm{FP}_3$, and have exponential, or polynomial Dehn functions of prescribed degree.

Groups with small Dehn functions and bipartite chord diagrams

We introduce a new invariant of bipartite chord diagrams and use it to construct the first examples of groups with Dehn function $n^2\log n$ and other small Dehn functions. Some of these groups have undecidable conjugacy problem.

Averaged Dehn Functions for Nilpotent Groups

Gromov proposed an averaged version of the Dehn function and claimed that in many cases it should be subasymptotic to the Dehn function. Using results on random walks in nilpotent groups, we confirm this claim for most nilpotent groups. In particular, if a nilpotent group satisfies the isoperimetric inequality $δ(l)<Cl^α$ for $α>2$ then it satisfies the averaged isoperimetric inequality $δ^{\text{avg}}(l)<C'l^{α/2}$. In the case of non-abelian free nilpotent groups, the bounds we give are asymptotically sharp.

群图的Dehn函数的布朗定理

A Brown Theorem for Dehn functions of graphs of groups

该研究针对在单连通CW复形上满足特定条件的群，证明了其Dehn函数的上界，给出了有限性性质布朗定理的Dehn函数类似结果，回答了Zaremsky的问题，并得到了高阶Dehn函数的对应结果。

Left-orderable fundamental group and Dehn surgery on twist knots

For any hyperbolic twist knot in the 3-sphere, we show that the resulting manifold by $r$-surgery on the knot has left-orderable fundamental group if the slope $r$ satisfies the inequality $0\le r \le 4$.

On cabled knots, Dehn surgery, and left-orderable fundamental groups

Previous work of the authors establishes a criterion on the fundamental group of a knot complement that determines when Dehn surgery on the knot will have a fundamental group that is not left-orderable. We provide a refinement of this criterion by introducing the notion of a decayed knot; it is shown that Dehn surgery on decayed knots produces surgery manifolds that have non-left-orderable fundamental group for all sufficiently positive surgeries. As an application, we prove that sufficiently positive cables of decayed knots are always decayed knots. These results mirror properties of L-space …

Left-orderable fundamental group and Dehn surgery on the knot $5_2$

We show that the resulting manifold by $r$-surgery on the knot $5_2$, which is the two-bridge knot corresponding to the rational number 3/7, has left-orderable fundamental group if the slope $r$ satisfies $0\le r\le 4$.

Dehn functions for mapping tori of amalgam of free groups

For an amalgam of two free groups and a particular kind of automorphism, we show that the Dehn function of the corresponding mapping torus is quadratic.

The Dehn functions of Stallings-Bieri groups

We show that the Stallings-Bieri groups, along with certain other Bestvina-Brady groups, have quadratic Dehn function.

Centralisers of Dehn twist automorphisms of free groups

We refine Cohen and Lustig's description of centralisers of Dehn twists of free groups. We show that the centraliser of a Dehn twist of a free group has a subgroup of finite index that has a finite classifying space. We describe an algorithm to find a presentation of the centraliser. We use this algorithm to give an explicit presentation for the centraliser of a Nielsen automorphism in Aut(F_n). This gives restrictions to actions of Aut(F_n) on CAT(0) spaces.

Homological and homotopical Dehn functions are different

The homological and homotopical Dehn functions are different ways of measuring the difficulty of filling a closed curve inside a group or a space. The homological Dehn function measures fillings of cycles by chains, while the homotopical Dehn function measures fillings of curves by disks. Since the two definitions involve different sorts of boundaries and fillings, there is no a priori relationship between the two functions, but prior to this work there were no known examples of finitely-presented groups for which the two functions differ. This paper gives the first such examples, constructed …

自由群外自同构群中的Dehn扭转动力学

Dynamics of Dehn Twists in the Outer Automorphism Group of a Free Group

研究有限生成非交换自由群外自同构群中的Dehn扭转，证明在兼容条件下，有限多个Dehn扭转的足够大幂次生成右角Artin群，并建立bigon-bihedron准则和交换条件。

Farrell-Jones via Dehn fillings

Following the approach of Dahmani, Guirardel and Osin, we extend the group theoretical Dehn filling theorem to show that the pre-images of infinite order elements have a certain structure of a free product. We then apply this result to show that groups hyperbolic relative to residually finite groups satisfying the Farrell-Jones conjecture, they satisfy the Farrell-Jones conjecture.

Left-orderable fundamental group and Dehn surgery on genus one two-bridge knots

For any hyperbolic genus one 2-bridge knot in the 3-sphere, we show that the resulting manifold by $r$-surgery on the knot has left-orderable fundamental group if the slope $r$ lies in some range which depends on the knot.

On a generalization of Dehn's algorithm

Viewing Dehn's algorithm as a rewriting system, we generalise to allow an alphabet containing letters which do not necessarily represent group elements. This extends the class of groups for which the algorithm solves the word problem to include nilpotent groups, many relatively hyperbolic groups including geometrically finite groups and fundamental groups of certain geometrically decomposable manifolds. The class has several nice closure properties. We also show that if a group has an infinite subgroup and one of exponential growth, and they commute, then it does not admit such an algorithm. W…

On mapping class group quotients by powers of Dehn twists and their representations

The aim of this paper is to survey some known results about mapping class group quotients by powers of Dehn twists, related to their finite dimensional representations and to state some open questions. One can construct finite quotients of them, out of representations with Zariski dense images into semisimple Lie groups. We show that, in genus 2, the Fibonacci TQFT representation is actually a specialization of the Jones representation. Eventually, we explain a method of Long and Moody which provides large families of mapping class group representations.

双曲群子群的 Dehn 函数的显式多项式上界

Explicit polynomial bounds on Dehn functions of subgroups of hyperbolic groups

本文证明 Brady 构造的双曲群非双曲有限呈示子群的 Dehn 函数具有显式多项式上界 $n^{96}$，并确定其万有覆盖 1-骨架的精确双曲性常数。

半单李群中的 Dehn 填充

Dehn filling in semisimple Lie groups

将Thurston双曲Dehn填充定理推广至任意秩半单李群，证明扩展几何有限子群的形变保持性质，并给出相对Anosov子群形变保持Anosov的准则，适用于凸射影几何高秩例子。

具有层次超几何群的漏斗空间，及其在德恩填充商中的应用

Cusped spaces for hierarchically hyperbolic groups, and applications to Dehn filling quotients

本文提出一种构造方法，同时产生相对超几何群的漏斗空间和准等距于泰克米勒度量的空间。通过此方法研究了各种群的德恩填充类似商，包括穿孔球的映射类群。特别地，证明了五孔球的映射类群（或四strand辫群）有无限超几何商，且不与其他球面映射类群（或辫群）的超几何商同构。

基于群论的Dehn填充视角：Property P猜想及其扩展

Group theoretic perspective on Dehn fillings: Property P conjecture and beyond

本文从群论角度提出研究Dehn填充的新视角，探讨Property P猜想及其扩展。

Thompson群V的Dehn函数至多为六次

The Dehn function of Thompson's group $V$ is at most sextic

本文将Thompson群V的Dehn函数上界从n¹¹改进为n⁶，同时通过显式van Kampen图探究其关系复杂性，补充了Thompson群Dehn函数的相关研究进展。

The Dehn function of Stallings' group

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.

Connectedness of Bowditch Boundary of Dehn Fillings

We study Dehn fillings of relatively hyperbolic group pairs $(Γ, ¶)$ and the persistence of connectedness of Bowditch boundary in sufficiently long Dehn fillings. We show that the restriction of peripheral subgroups to virtually polycyclic subgroups is not needed.

Left-orderable fundamental groups and Dehn surgery

There are various results that frame left-orderability of a group as a geometric property. Indeed, the fundamental group of a 3-manifold is left-orderable whenever the first Betti number is positive; in the case that the first Betti number is zero this property is closely tied to the existence of certain nice foliations. As a result, many large classes of 3-manifolds, including knot complements, are known to have left-orderable fundamental group. However, though the complement of a knot has left-orderable fundamental group, the result of Dehn surgery is a closed 3-manifold that need not have t…

Bestvina-Brady群的Dehn函数的完全分类

Complete classification of the Dehn functions of Bestvina-Brady groups

本文证明了每个有限表示的Bestvina-Brady群的Dehn函数增长为线性、二次、三次或四次多项式，并给出了确定多项式次数的显式判据，进而识别了阻碍某些Bestvina-Brady群具有CAT(0)结构的障碍。

Dehn 拟群的展示

Presentations of Dehn quandles

本文提出两种利用底层群展示构造 Dehn 拟群显式展示的方法，并给出球面 Artin 群、曲面群和映射类群等实例。

The Dehn function of Sp(2n;Z)

Gromov conjectured that any irreducible lattice in a symmetric space of rank at least 3 should have at most polynomial Dehn function. We prove that the lattice Sp(2p;Z) has quadratic Dehn function when p is at least 5. By results of Broaddus, Farb, and Putman, this implies that the Torelli group in large genus is at most exponentially distorted.

Dehn filling Dehn twists

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…

Identifying Dehn Functions of Bestvina--Brady Groups From Their Defining Graphs

Let $Γ$ be a finite simplicial graph such that the flag complex on $Γ$ is a $2$-dimensional triangulated disk. We show that with some assumptions, the Dehn function of the associated Bestvina--Brady group is either quadratic, cubic, or quartic. Furthermore, we can identify the Dehn function from the defining graph $Γ$.

On squares of Dehn twists about non-separating curves of a non-orientable closed surface

The level $2$ mapping class group of an orientable closed surface can be generated by squares of Dehn twists about non-separating curves. On the other hand, the level $2$ mapping class group $\mathcal{M}_2(N_g)$ of a non-orientable closed surface $N_g$ can not be generated by only Dehn twists, and so it can not be generated by squares of Dehn twists about non-separating curves. In this paper, we prove that the Dehn twist subgroup of $\mathcal{M}_2(N_g)$ can not be generated by squares of Dehn twists about non-separating curves either. As an application, we give a finite generating set for the …

Torsion Subgroups of Groups with Cubic Dehn Function

We construct the first examples of finitely presented groups with cubic Dehn function containing a finitely generated infinite torsion subgroup. Moreover, we show that any infinite free Burnside group with sufficiently large odd exponent can be embedded as a subgroup of a finitely presented group with cubic Dehn function.

Deciding Isomorphy using Dehn fillings, the splitting case

We solve Dehn's isomorphism problem for virtually torsion-free relatively hyperbolic groups with nilpotent parabolic subgroups. We do so by reducing the isomorphism problem to three algorithmic problems in the parabolic subgroups, namely the isomorphism problem, separation of torsion (in their outer automorphism groups) by congruences, and the mixed Whitehead problem, an automorphism group orbit problem. The first step of the reduction is to compute canonical JSJ decompositions. Dehn fillings and the given solutions of the algorithmic problems in the parabolic groups are then used to decide if…

关于多项循环群族的 Dehn 函数

On Dehn functions for family of polycyclic groups

本文引入了有限群族的 Dehn 函数概念，研究了特定有限多项循环群族的 Dehn 函数，并探讨了有限单循环群族的球面 Dehn 函数和平均 Dehn 函数。

The Dehn function of Baumslag's Metabelian Group

Baumslag's group is a finitely presented metabelian group with a Z \wr Z subgroup. There is an analogue with an additional torsion relation in which this subgroup becomes C_m \wr Z. We prove that Baumslag's group has an exponential Dehn function. This contrasts with the torsion analogues which have quadratic Dehn functions. v.2: version for publication, incorporating referee's suggestions.

Generalized Dehn Functions II

We establish the existence, finiteness, and uniqueness up to scaling of various isoperimetric profiles of a group, in all dimensions. We also show that these profiles all coincide in dimensions 4 and higher; in particular, the nth Dehn function is equal to FV^{n+1} for n at least 3. Even for dimension 3, there is significant overlap. When a group has decidable word problem, this has mild consequences for the growth rates of its profiles. We also establish a metric analogue for highly connected Riemannian manifolds.

Left-orderability and exceptional Dehn surgery on two-bridge knots

We show that any exceptional non-trivial Dehn surgery on a hyperbolic two-bridge knot yields a 3-manifold whose fundamental group is left-orderable. This gives a new supporting evidence for a conjecture of Boyer, Gordon and Watson.

Dehn Function of Finitely Presented Metabelian Groups

In this paper, we compute an upper bound for the Dehn function of a finitely presented metabelian group. In addition, we prove that the same upper bound works for the relative Dehn function of a finitely generated metabelian group. We also show that every wreath product of a free abelian group of finite rank with a finitely generated abelian group can be embedded into a metabelian group with exponential Dehn function.