arXiv research

The paper presents methods to write presentations for Dehn quandles.

Goeritz and Seifert matrices derived from Dehn presentations.

Presentations for involutions on non-orientable surfaces up to genus 5.

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

Study infinite group presentations and their Dehn functions.

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

We classify Dehn functions of Bestvina-Brady groups.

The paper shows how coarse embeddings affect homological Dehn functions.

Explicit polynomial bound found for subgroup Dehn function.

New spectral Dehn function characterizes word-hyperbolic groups.

A version of Dehn's algorithm for simple diagrams on a once punctured surface representing simple diagrams on a closed surface is presented

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

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 ex…

The paper finds Artin presentations for the trivial group and identifies hyperbolic 3-braids.

Classifies periodic diffeomorphisms and hyperelliptic involutions on surfaces.

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 give a method to compute presentations of saturated cluster modular groups. Using this, we obtain finite presentations of the saturated cluster modular groups of finite mutation type $X_6$ and $X_7$. We verify that the cluster modular groups of finite mutation type $\widetilde{E}_6$, $\widetilde{E}_7$, $\widetilde{E…

In this paper, we present a new approach to the construction of Einstein metrics by a generalization of Thurston's Dehn filling. In particular in dimension 3, we will obtain an analytic proof of Thurston's result.

This article is solicited by C.\ Adams for a special issue of {\it Chaos, Solitons and Fractals\/} devoted to knot theory and its applications. We present some recent results about Dehn surgeries on arborescent knots and links.

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.

The positive Dehn twist expressions for the generalizations of the new involutions described in math.GT/0404310 are presented. The homeomorphism types of the Lefschetz fibrations that they define are determined for several examples.

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.

This paper provides an infinite presentation for a subgroup of mapping class groups of non-orientable surfaces.

Finite set of Dehn twists describes genus-2 Goeritz group elements.

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…

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…

A positive Dehn twist product for a $\mathbb{Z}_3$ action with $g+2$ fixed points on the 2-dimensional closed, compact, oriented surface $Σ_g$ is presented. The homeomorphism invariants of the resulting symplectic 4-manifolds are computed.

Using the works of Gervais, Harer, Hatcher and Thurston and others, we show that the mapping class group of a compact orientable surface has a presentation so that the generators are the set of all Dehn twists and the relations are supported in subsurfaces homeomorphic to the one-holed torus or the four-holed sphere. I…

The paper explores group presentations for links in thickened surfaces, proving their relationship and introducing new invariants.

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…

Study of Dehn twists on a disc with 3 points, solving conjugacy problem.

We provide a calculus for the presentation of closed 3-manifolds via nullhomotopic filling Dehn spheres and we use it to define an invariant of closed 3-manifolds by applying the state-sum machinery. As a potential application of this invariant, we show how to get lower bounds for the Matveev complexity of P2-irreducib…

Let N_{g,s} denote the nonorientable surface of genus g with s boundary components. Recently Paris and Szepietowski obtained an explicit finite presentation for the mapping class group M(N_{g,s}) of the surface N_{g,s}, where s\in{0,1} and g+s>3. Following this work we obtain a finite presentation for the mapping class…

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

Positive Dehn twist products for some elements of finite order in the mapping class group of a 2-dimensional closed, compact, oriented surface $Σ_g$, which are rotations of $Σ_g$ through $2π/p$, are presented. The homeomorphism invariants of the resulting simply connected symplectic 4- manifolds are computed.

New groups with distinct Dehn functions and properties.

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

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…

Study uses Dynnikov coordinates to analyze actions of Dehn twists on a thrice-punctured disc.

Extends classification of hyperbolic Dehn fillings in quadratic fields.

This paper computes fixed point Floer cohomology for Dehn twists on surfaces.

The paper connects hyperbolic Dehn surgery and Higgs bundles to construct model objects in representation varieties.

A persistent lamination for a knot K is an essential lamination in the complement of the K, which remains essential after every non-trivial Dehn surgery along K. In particular, this implies that all of the Dehn surgery manifolds have universal cover R^3. This paper presents a method for building tangles T with the prop…

Refined 3D index uses surgery and gradings to distinguish 3-manifolds.

The paper proves left-orderability for certain Dehn fillings of pseudo-Anosov mapping tori.

In this note we give presentations of all finite subgroups of the mapping class group of a closed surface of genus 2 by the Humphries generators up to conjugacy.

We give an infinite presentation for the mapping class group of a non-orientable surface. The generating set consists of all Dehn twists and all crosscap pushing maps along simple loops.

We give a finite presentation of the mapping class group of an oriented (possibly bounded) surface of genus greater or equal than 1, considering Dehn twists on a very simple set of curves.