Jose Burillo, Jennifer Taback
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.
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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…
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 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.
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 introduce a number of new tools for the study of relatively hyperbolic groups. First, given a relatively hyperbolic group G, we construct a nice combinatorial Gromov hyperbolic model space acted on properly by G, which reflects the relative hyperbolicity of G in many natural ways. Second, we construct two useful bicombings on this space. The first of these, "preferred paths", is combinatorial in nature and allows us to define the second, a relatively hyperbolic version of a construction of Mineyev. As an application, we prove a group-theoretic analog of the Gromov-Thurston 2πTheorem in the …
The aim of this paper is to demonstrate that very many Dehn fillings on a cusped hyperbolic 3-manifold yield a 3-manifold which is irreducible, atoroidal and not Seifert fibred, and which has infinite, word hyperbolic fundamental group. We establish an extension of the Thurston-Gromov $2π$ theorem by showing that if each filling slope has length more than six, then the resulting 3-manifold has all the above properties. We also give a combinatorial version of the $2π$ theorem which relates to angled ideal triangulations. We apply these techniques by studying surgery along alternating links.
Surfaces proper homotopy equivalent to graphs and their Dehn-Nielsen-Baer maps
研究可能为无限型且边界非紧的第二可数可定向曲面与图恰当同伦等价的充要条件，探索相关映射类群间诱导映射的基本性质，以此为无限型曲面背景下德恩 - 尼尔森 - 贝尔类似物奠定基础。
Cone-equivalent nilpotent groups with different Dehn functions
本研究构造了一族k步幂零李群，其与对应Carnot分次群渐近锥双李普希茨等价但Dehn函数增长阶不同，揭示了三类新现象并回答了相关领域的公开问题。
Quasi-isometry invariance of relative filling functions
本研究证明有限生成群与子群配对的相对Dehn函数具有拟等距不变性，揭示拟等距对几乎反正规性、锥化Cayley图精细性的保持作用，明确组合Dehn函数良定义的等价条件，给出双曲嵌入子群相关结论及Baumslag-Solitar群的特例结果。
On the Dehn functions of central products of nilpotent groups
本文确定了多类幂零李群中心积的Dehn函数，证实了相关猜想，并构造了无限族群使Cornulier的界渐近最优。
A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups
本研究提出基于双曲单纯复形作用的层级双曲群组合判据，证明映射类群模Dehn扭大幂次的商群为层级双曲群，构造非初等双曲商群并将相关有限商问题归约为特定双曲群的剩余有限性问题。
We prove that when n>=5, the Dehn function of SL(n,Z) is at most quartic. The proof involves decomposing a disc in SL(n,R)/SO(n) into a quadratic number of loops in generalized Siegel sets. By mapping these loops 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.
Sublinear Bilipschitz Equivalence and the Quasiisometric Classification of Solvable Lie Groups
本文通过亚线性双 Lipschitz 等价理论区分可解李群的准等距类别，揭示了高维非单模可解群的不可数多个准等距类。
Combinatorial equivalence of separable elements in types $A$ and $B$
研究$A$型和$B$型可分元素的组合等价性，构造对称群与超八面体群间的双射，证明降序多项式$\gamma$-正性并推导秩生成函数显式公式。
An equivalence relation in the symmetric group, where is a positive integer has been considered. An algorithm for calculation of the number of the equivalence classes by this relation for arbitrary integer has been described.
Filling functions are asymptotic invariants of finitely presentable groups; the seminal work on the subject is by M.Gromov. They record features of combinatorial homotopy discs (van Kampen diagrams) filling loops in Cayley 2-complexes. Examples are the Dehn (or isoperimetric) function, the filling length function and the intrinsic diameter (or isodiametric) function. We discuss filling functions from geometric, combinatorial and computational points of view, we survey their interrelationships, and we sketch their roles in the studies of nilpotent groups, hyperbolic groups and asymptotic cones.…
The Space of Actions, Partition Metric and Combinatorial Rigidity
本文在 d 生成群的作用空间上构造自然伪度量，刻画 Kechris 弱等价类并得到紧度量空间，且借助符号动力学与超积将结果推广至酉表示。
The existence and uniqueness (up to equivalence) of code loops was first established by R. Griess. Nevertheless, the explicit construction of code loops remained open until T. Hsu introduced the notion of symplectic cubic spaces and their Frattini extensions, and pointed out how the construction of code loops followed from the (purely combinatorial) result of O. Chein and E. Goodaire. Within this paper, we focus on their combinatorial construction and prove a more general result using the language of derived forms.
A graph is Helly if every family of pairwise intersecting combinatorial balls has a nonempty intersection. We show that weak Garside groups of finite type and FC-type Artin groups are Helly, that is, they act geometrically on Helly graphs. In particular, such groups act geometrically on spaces with convex geodesic bicombing, equipping them with a nonpositive-curvature-like structure. That structure has many properties of a CAT(0) structure and, additionally, it has a combinatorial flavor implying biautomaticity. As immediate consequences we obtain new results for FC-type Artin groups (in parti…
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