Josh Barnard, Noel Brady, Pallavi Dani
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.
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For n > 2, the Dehn functions of Aut(F_n) and Out(F_n) are exponential. Hatcher and Vogtmann proved that they are at most exponential, and the complementary lower bound in the case n=3 was established by Bridson and Vogtmann. Handel and Mosher completed the proof by reducing the lower bound for n>4 to the case n=3. In this note we give a shorter, more direct proof of this last reduction.
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 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.
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.
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…
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.
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.
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.
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。
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.
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.
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 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 study the Dehn function of connected Lie groups. We show that this function is always exponential or polynomially bounded, according to the geometry of weights and of the 2-cohomology of their Lie algebras. Our work, which also addresses algebraic groups over local fields, uses and extends Abels' theory of multiamalgams of graded Lie algebras, in order to provide workable presentations of these groups.
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.
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。
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 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…
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 …
Dehn functions: computations, lower bounds, and the quasiisometric rigidity of $\rm{Sol}_5$
本文利用次线性双李普希茨收缩在完全可解李群中建立扭曲估计以给出Dehn函数新下界，并计算了5维内指数增长单连通可解李群的Dehn函数与渐近锥维数，证明了$\ m{Sol}_5$及其格的拟等距刚性。
McCarthy's Theorem for the mapping class group of a closed hyperbolic surface states that for any two mapping classes $σ,τ\in \mathrm{Mod}(S)$ there is some power $N$ such that the group $\langle σ^N,τ^N\rangle$ is either free of rank two or abelian, and gives a geometric criterion for the dichotomy. The analogous statement is false in linear groups, and unresolved for outer automorphisms of a free group. Several analogs are known for exponentially growing outer automorphisms satisfying various technical hypothesis. In this article we prove an analogous statement when $σ$ and $τ$ are linearly …
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.
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。
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。
A discrete group with word-length (G,L) is B-isocohomological for a bounding classes B if the comparison map from B-bounded cohomology to ordinary cohomology (with complex coefficients) is an isomorphism; it is strongly B-isocohomological if the same is true with arbitrary coefficients. In this paper we establish some basic conditions guaranteeing strong B-isocohomologicality. In particular, we show strong B-isocohomologicality for an $FP^{\infty}$ group G if all of the weighted G-sensitive Dehn functions are B-bounded. Such groups include all B-asynchronously combable groups; moreover, the cl…
Weak commutativity, virtually nilpotent groups, and Dehn functions
本文研究弱交换构造 $\mathfrak{X}(G)$，通过分析正规交换子群模结构，证明其保持虚拟幂零、Engel 与增长类型，并刻画字问题及 Dehn 函数下界。
Conjugator Length in Finitely Presented Groups
研究有限呈现群的共轭长度函数，证明其能实现所有可作为德恩函数的函数，并构造出渐近等同于n^α的群。
This is the first of two papers devoted to connections between asymptotic functions of groups and computational complexity. One of the main results of this paper states that if for every $m$ the first $m$ digits of a real number $α\ge 4$ are computable in time $\le C2^{2^{Cm}}$ for some constant $C>0$ then $n^α$ is equivalent (``big O'') to the Dehn function of a finitely presented group. The smallest isodiametric function of this group is $n^{3/4α}$. On the other hand if $n^α$ is equivalent to the Dehn function of a finitely presented group then the first $m$ digits of $α$ are computable in t…
Conjugator Lengths and Isoperimetric functions
该研究构造了有限表示群，揭示共轭子长度函数与Dehn函数这两个不变量具有强独立性，还给出了共轭子长度函数递归但字、共轭问题均不可判定的首个例子。
We construct 2-dimensional CAT(-1) groups which contain free subgroups with arbitrary iterated exponential distortion, and with distortion higher than any iterated exponential.
The Dehn function and its higher-dimensional generalizations measure the difficulty of filling a sphere in a space by a ball. In nonpositively curved spaces, one can construct fillings using geodesics, but fillings become more complicated in subsets of nonpositively curved spaces, such as lattices in symmetric spaces. In this paper, we prove sharp filling inequalities for (arithmetic) lattices in higher rank semisimple Lie groups. When $n$ is less than the rank of the associated symmetric space, we show that the $n$-dimensional filling volume function of the lattice grows at the same rate as t…
Wreath-like products of groups and their von Neumann algebras I: $W^\ast$-superrigidity
本文定义类圈积群，证明一类具有 Kazhdan 性质 (T) 的类圈积可由其群 von Neumann 代数唯一恢复同构类，并给出 $2^{\aleph_0}$ 个此类 $W^*$-超刚性群。
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