Jennifer Taback
We show that PSL(2,Z[1/p]) admits a combing with bounded asynchronous width, and use this combing to show that PSL(2,Z[1/p]) has an exponential Dehn function. As a corollary, PSL(2,Z[1/p]) has solvable word problem and is not an automatic group.
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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。
We prove that the Dehn function (that is, the smallest isoperimetric function) of the Richard Thompson's group F is quadratic.
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.
We construct a finitely presented group with undecidable word problem and with Dehn function bounded by a quadratic function on an infinite set of positive integers.
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.
We show that the Stallings-Bieri groups, along with certain other Bestvina-Brady groups, have quadratic Dehn function.
We obtain an upper bound for relative Dehn functions of amalgamated products and HNN--extensions with respect to certain collections of subgroups. Our main results generalize the combination theorems for relatively hyperbolic groups proved by Dahmani.
We prove that the Word problem in the Baumslag group G(1,2) which has a non-elementary Dehn function is decidable in polynomial time.
We show the connection between the relative Dehn function of a finitely generated metabelian group and the distortion function of a corresponding subgroup in the wreath product of two free abelian groups of finite rank. Further, we show that if a finitely generated metabelian group $G$ is an extension of an abelian group by $\mathbb Z$ the relative Dehn function of $G$ is polynomially bounded. Therefore, if $G$ is finitely presented, the Dehn function is bounded above by the exponential function up to equivalence.
We prove that groups in a certain class of metabelian locally compact groups, have quadratic Dehn function. As an application, we embed the solvable Baumslag-Solitar groups into finitely presented metabelian groups with quadratic Dehn function. Also, we prove that Baumslag's finitely presented metabelian groups, in which the lamplighter groups embed, have quadratic Dehn function.
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.
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.
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.
We construct the first examples of finitely presented groups with quadratic Dehn function containing a finitely generated infinite torsion subgroup. These examples are "optimal" in the sense that the Dehn function of any such finitely presented group must be at least quadratic. Moreover, we show that for any $n\geq2^{48}$ such that $n$ is either odd or divisible by $2^9$, any infinite free Burnside group with exponent $n$ is a quasi-isometrically embedded subgroup of a finitely presented group with quadratic Dehn function satisfying the Congruence Extension Property.
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 $Γ$.
Thompson's group T has quadratic Dehn function
本文证明了Thompson群T及所有Higman-Thompson群T_n的Dehn函数为二次的，这是对其几何性质的重要刻画。
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.
We construct a finitely presented group with quadratic Dehn function and undecidable conjugacy problem. This solves E. Rips' problem formulated in 1992. v2: misprints corrected. v3: lemmas 4.7, 4.10 corrected, more misprints fixed.
On the one hand, it is well known that the only subquadratic Dehn function of finitely presented groups is the linear one. On the other hand there is a huge class of Dehn functions $d(n)$ with growth at least $n^4$ (essentially all possible such Dehn functions) constructed in \cite{SBR} and based on the time functions of Turing machines and S-machines. The class of Dehn functions $n^α$ with $α\in (2; 4)$ remained more mysterious even though it has attracted quite a bit of attention (see, for example, \cite{BB}). We fill the gap obtaining Dehn functions of the form $n^α$ (and much more) for all…
We show that for each positive integer $k$ there exist right-angled Artin groups containing free-by-cyclic subgroups whose monodromy automorphisms grow as $n^k$. As a consequence we produce examples of right-angled Artin groups containing finitely presented subgroups whose Dehn functions grow as $n^{k+2}$.
The Dehn function for Palindromic Sub-Group of $Aut(F_n)$
本文证明了回文自同构群 $ΠA(F_n)$ 的 Dehn 函数是指数级的，这是对该群几何性质的重要刻画。
We introduce two new types of Dehn functions of group presentations which seem more suitable (than the standard Dehn function) for infinite group presentations and prove the fundamental equivalence between the solvability of the word problem for a group presentation defined by a decidable set of defining words and the property of being computable for one of the newly introduced functions (this equivalence fails for the standard Dehn function). Elaborating on this equivalence and making use of this function, we obtain a characterization of finitely generated groups for which the word problem ca…
V.S.Guba had proved that the R.Thompson group $T$ satisfies polynomial isoperimetric inequality and $Φ_T(n)\preceq n^7$, where $Φ_T$ is the Dehn function of group $T$. In this paper, we show that $Φ_T(n)\preceq n^5$.
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.
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。
Dehn functions of coabelian subgroups of direct products of groups
本研究提出计算群直积的余阿贝尔子群精确Dehn函数的新方法，推广了前人相关结果，并将其应用于多类群的相关研究中。
In the space of marked group, we suppose that a sequence $(G_i, X_i)$ converges to $(G,X)$, where $G$ is finitely presented. We obtain an inequality which connects Dehn functions of $G_i$s and $G$. As a result, we show that if a sequence $(K, X_i)$ converges to a hyperbolic marked group, then $K$ is already hyperbolic.
We prove that Abels' group over an arbitrary nondiscrete locally compact field has a quadratic Dehn function. As applications, we exhibit connected Lie groups and polycyclic groups whose asymptotic cones have uncountable abelian fundamental group. We also obtain, from the case of finite characteristic, uncountably many non-quasi-isometric finitely generated solvable groups, as well as peculiar examples of fundamental groups of asymptotic cones.
Finitely presented simple groups with at least exponential Dehn function
本研究借助经仔细校准的Baumslag-Solitar群自相似表示，从Röver-Nekrashevych群出发，构造出首批已知的Dehn函数至少为指数级的有限表现单群实例。
We construct a finitely presented group $G$ with non-quadratic Dehn function $f$ majorizable by a quadratic function on arbitrary long intervals.
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…
Polynomial homological Dehn functions from non-proper actions
建立同调代数框架证明群的高阶同调Dehn函数的多项式性，并给出多项式Dehn函数的组合定理。
We prove super-quadratic lower bounds for the growth of the filling area function of a certain class of Carnot groups. This class contains groups for which it is known that their Dehn function grows no faster than $n^2\log n$. We therefore obtain the existence of (finitely generated) nilpotent groups whose Dehn functions do not have exactly polynomial growth and we thus answer a well-known question about the possible growth rate of Dehn functions of nilpotent groups.
We prove that if a finitely presented group acts properly discontinuously, cocompactly and by isometries on a simply connected Riemannian manifold, then the Dehn function of the group and the corresponding filling function of the manifold are equivalent, in a sense described below.
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.
We construct examples of finitely generated decidable group presentations that satisfy certain combinations of solvability for the word problem, solvability for the bounded word problem, and computablity for the Dehn function. We prove that no finitely generated decidable presentations exist satisfying the combinations for which we do not provide examples. The presentations we construct are also minimal. These constructions answer an open question asked by R.I. Grigorchuk and S.V. Ivanov. Our approach uses machinery developed by Birget, Ol'shanskii, Rips, and Sapir for constructing finite grou…
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。
Quasi-isometric Higman embeddings and the Dehn function
本文证明满足特定计算枚举条件的有限生成群可拟等距嵌入到 Dehn 函数受计算复杂度控制的有限呈现群中，推广了自由 Burnside 群嵌入结果，并改进了字问题 NP 性与多项式 Dehn 函数嵌入的经典刻画。
Dehn functions of mapping tori of right-angled Artin groups
本文研究直角 Artin 群在自同构下的代数映射环面，并对所有 3 生成元直角 Artin 群及 F_k × F_l（k,l ≥ 2）按自同构 Φ 分类其 Dehn 函数。
This paper shows that a finitely presented monoid with linear Dehn function need not have a regular cross-section, strengthening the previously-known result that such a monoid need not be presented by a finite complete string rewriting system, and contrasting the fact that finitely presented groups with linear Dehn function always have regular cross-sections.
We produce examples of groups of type F_3 with 2-dimensional Dehn functions of the form exp^n(x) (a tower of exponentials of height n), where n is any natural number.
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 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 …
Nilpotent groups with Dehn function $n^2 \log n$
该研究构造了具有Dehn函数$n^2 \log n$的2步Carnot群，改进了Wenger的下界，得到首个幂零群中不按$n^\alpha$（$\alpha \geq 1$）增长的Dehn函数的精确计算结果。
From Ideal Membership Problem for polynomial rings to Dehn Functions of Metabelian Groups
本文定义整数系数Laurent多项式环中理想成员问题的复杂性函数，并将其与亚交换群的Dehn函数相联系，为构造超指数Dehn函数的亚交换群提供途径。
In 2013, Kharlampovich, Myasnikov, and Sapir constructed the first examples of finitely presented residually finite groups with large Dehn functions. Given any recursive function $f$, they produce a finitely presented residually finite group with Dehn function dominating $f$. There are no known elementary examples of finitely presented residually finite groups with super-exponential Dehn function. Dison and Riley's hydra groups can be used to construct a sequence of groups for which the Dehn function of the k-th group is equivalent to the k-th Ackermann function. Kharlampovich, Myasnikov, and …
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.
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.
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 …
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.
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.
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。
Random Dehn function of groups
本文研究群的随机Dehn函数概念，计算了Dehn函数至多为多项式的有限表示非紧双曲群的渐近上界，证实了Gromov的直觉并证明其随机Dehn函数至多为二次函数。
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.
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.
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.
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.
While Dehn functions, D(n), of finitely presented groups are very well studied in the literature, mean Dehn functions are much less considered. M. Gromov introduced the notion of mean Dehn function of a group, $D_{mean}(n)$, suggesting that in many cases it should grow much more slowly than the Dehn function itself. Using only elementary counting methods, this paper presents some computations pointing into this direction. Particularizing them to the case of any finite presentation of a finitely generated abelian group (for which it is well known that $D(n)\sim n^2$ except in the 1-dimensional …
We give an infinite family of monoids $Π_N$ (for $N=2, 3, \dots$), each with a single defining relation of the form $bUa = a$, such that the Dehn function of $Π_N$ is at least exponential. More precisely, we prove that the Dehn function $\partial_N(n)$ of $Π_N$ satisfies $\partial_N(n) \succeq N^{n/4}$. This answers negatively a question posed by Cain & Maltcev in 2013 on whether every monoid defined by a single relation of the form $bUa=a$ has quadratic Dehn function. Finally, by using the decidability of the rational subset membership problem in the metabelian Baumslag--Solitar groups $\oper…
A semigroup with linearithmic Dehn function
本文构造了一个 Dehn 函数为 $n \log n$ 的半群，填补了群论中线性与二次函数之间的渐近间隙，并探讨了 semi-Thue 系统推导复杂度与对称化半群 Dehn 函数之间的非普遍联系。
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