arXiv research

Study on homological Dehn functions of groups of type $FP_2$.

Precise computations of Dehn functions for subgroups of free group products.

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 a…

New CAT(0) groups show superexponential subgroup Dehn functions.

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 functio…

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.

The study bounds Dehn functions of coabelian subgroups using a second BNSR invariant.

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 classify Dehn functions of Bestvina-Brady groups.

Explicit polynomial bound found for subgroup Dehn function.

Researchers determine Dehn functions of specific nilpotent groups.

New groups with distinct Dehn functions and properties.

New spectral Dehn function characterizes word-hyperbolic groups.

The paper shows how coarse embeddings affect homological Dehn functions.

We construct a finitely presented group $G$ with non-quadratic Dehn function $f$ majorizable by a quadratic function on arbitrary long intervals.

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 invo…

Metric spaces with upper curvature bounds have controlled Dehn functions.

The paper proves that relative Dehn functions are invariant under quasi-isometry.

Study infinite group presentations and their Dehn functions.

We give a lower estimate of the framing function of knots, and prove a strengthened version of Dehn's lemma conjectured by Greene-Wiest.

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 automorphis…

We prove that any proper, geodesic metric space whose Dehn function grows asymptotically like the Euclidean one has asymptotic cones which are non-positively curved in the sense of Alexandrov, thus are ${\rm CAT}(0)$. This is new already in the setting of Riemannian manifolds and establishes in particular the borderlin…

Stability of Dehn functions proven for ultralimits of Sobolev maps.

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.

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.

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…

Dehn surgery homeomorphic pairs contradict a conjecture.

In this paper it is proved that if a finitely presented group acts properly discontinuously, cocompactly and by isometries on a simply connected Riemannian manifold, then the two Dehn functions, of the group and the manifold, respectively, are equivalent.

We prove a sharp estimate on the expected value of the integral of the index of a simple random walk on the square or triangular lattice. This gives new lower bounds on the averaged Dehn function, which measures the expected area needed to fill a random curve with a disc.

Researchers find a way to bound the complexity of certain subgroup geometric invariants.

The k-dimensional Dehn (or isoperimetric) function of a group bounds the volume of efficient ball-fillings of k-spheres mapped into k-connected spaces on which the group acts properly and cocompactly; the bound is given as a function of the volume of the sphere. We advance significantly the observed range of behavior f…

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 multiamal…

The paper examines slopes and their norms in exceptional Dehn fillings.

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 tec…

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 h…

Let $X=S\times E \times B$ be the metric product of a symmetric space $S$ of noncompact type, a Euclidean space $E$ and a product $B$ of Euclidean buildings. Let $Γ$ be a discrete group acting isometrically and cocompactly on $X$. We determine a family of quasi-isometry invariants for such $Γ$, namely the $k$-dimension…

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 polyn…

New rigidity result for hyperbolic surfaces based on curve lengths.

The study bounds slopes for Dehn fillings of two-bridge knots with hyperbolic representations.

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 …

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 signi…

Proves formula for 3D index change with Dehn filling.

Dehn quandles of groups and surfaces unify various quandle constructions.

The paper presents methods to write presentations for Dehn quandles.

Study finds the order of Dehn twists in various groups.

Study shows connectedness of Bowditch boundary persists in long Dehn fillings.

Four-dimensional Einstein Dehn filling is impossible.