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Let $$\Gamma $$ be a finite simplicial graph such that the flag complex on $$\Gamma $$ 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 $$\Gamma $$ .

It is undecidable in general whether a given finitely presented group is word hyperbolic. We use the concept of pregroups, introduced by Stallings, to define a new class of van Kampen diagrams, which represent groups as quotients of virtually free groups. We then present a polynomial-time procedure which analyses these diagrams, and either returns an explicit linear Dehn function for the presentation, or returns fail, together with its reasons for failure. Furthermore, if our procedure succeeds we are often able to produce in polynomial time a word problem solver for the presentation that runs in linear time. Our algorithms have been implemented, and are often many orders of magnitude faster than KBMAG, the only comparable publicly available software., Comment: To appear in Journal of Symbolic Computation

For every pair of positive integers $p > q$ we construct a one-relator group $R_{p,q}$ whose Dehn function is $\simeq n^{2 \alpha}$ where $\alpha = \log_2(2p / q)$. The group $R_{p,q}$ has no subgroup isomorphic to a Baumslag-Solitar group $BS(m,n)$ with $m \neq \pm n$, but is not automatic, not CAT(0), and cannot act freely on a CAT(0) cube complex. This answers a long-standing question on the automaticity of one-relator groups and gives counterexamples to a conjecture of Wise., Comment: 6 pages, 1 figure; v3 final version to appear in Proceedings of the American Mathematical Society; v2 correct remark about residual finiteness

We investigate groups whose Cayley graphs have poor\-ly connected subgraphs. We prove that a finitely generated group has bounded separation in the sense of Benjamini--Schramm--Tim\'ar if and only if it is virtually free. We then prove a gap theorem for connectivity of finitely presented groups, and prove that there is no comparable theorem for all finitely generated groups. Finally, we formulate a connectivity version of the conjecture that every group of type $F$ with no Baumslag-Solitar subgroup is hyperbolic, and prove it for groups with at most quadratic Dehn function., Comment: 14 pages. Changes to v2: Proof of the Theorem 1.2 shortened, Theorem 1.4 added completing the no-gap result outlined in v1

We consider the word search problem in the Baumslag-Gersten group GB. We show that the parameterized complexity of this problem, where the area of van Kampen diagram serves as a parameter, is polynomial in the length of the input and the parameter. This contrasts the well-known result that the Dehn function and the time complexity of the word search problem in GB are non-elementary.

It is Thurston's result that for a hyperbolic knot $K$ in $S^{3}$, almost all Dehn fillings on its complement result in hyperbolic 3-manifolds except some exceptional cases. So almost all produced 3-manifolds have the same geometry. It is known that its complement in $S^{3}$, denoted by $E(K)$, admits a Heegaard splitting. Then it is expected that there is a similar result on Heegaard distance for Dehn fillings.In this paper, Dehn fillings on genus two Heegaard splittings are studied. More precisely, we prove that if the distance of a given genus two Heegaard splitting of $E(K)$ is at least 3, then for any two degenerating slopes on $\partial~E(K)$, there is a universal bound of their distance in the curve complex of $\partial~E(K)$.

Gromov conjectured that any irreducible lattice in a symmetric space of rank at least [Formula: see text] should have at most polynomial Dehn function. We prove that the lattice [Formula: see text] has quadratic Dehn function when [Formula: see text]. By results of Broaddus, Farb, and Putman, this implies that the Torelli group in large genus is at most exponentially distorted.

We establish compact presentability, i.e. the locally compact version of finite presentability, for an infinite family of tree almost automorphism groups. Examples covered by our results include Neretin's group of spheromorphisms, as well as the topologically simple group containing the profinite completion of the Grigorchuk group constructed by Barnea, Ershov and Weigel. We additionally obtain an upper bound on the Dehn function of these groups in terms of the Dehn function of an embedded Higman-Thompson group. This, combined with a result of Guba, implies that the Dehn function of the Neretin group of the regular trivalent tree is polynomially bounded., The results are extended to some almost automorphism groups of trees associated with closed regular branch groups. In particular we prove that the simple group (containing the profinite completion of the Grigorchuk group) constructed by Barnea, Ershov and Weigel, is compactly presented

We show that the Stallings–Bieri groups, along with certain other Bestvina–Brady groups, have quadratic Dehn function.

In this paper, we will show that for any link L in a closed orientable 3-manifold M, infinitely many hyperbolic 3-manifolds are obtained from M by Dehn surgeries so that each of them admits an orientation-preserving smooth finite cyclic group action with fixed point set L.

This short survey deals with the large-scale geometry of solvable groups. Instead of giving a global overview of this wide subject, we chose to focus on three aspects which illustrate the broad diversity of methods employed in this subject. The first one has probabilistic and analytic flavors, the second is related to cohomological properties of unitary representations, while the third one deals with the Dehn function. To keep the exposition concrete, we discuss lots of examples, mostly among solvable linear groups.

We introduce the notion of metrically systolic simplicial complexes. We study geometric and large-scale properties of such complexes and of groups acting on them geometrically. We show that all two-dimensional Artin groups act geometrically on metrically systolic complexes. As direct corollaries we obtain new results on two-dimensional Artin groups and all their finitely presented subgroups: we prove that the Conjugacy Problem is solvable, and that the Dehn function is quadratic. We also show several large-scale features of finitely presented subgroups of two-dimensional Artin groups, lying background for further studies concerning their quasi-isometric rigidity., Comment: final preprint version, to appear in Math. Ann

We construct a finitely presented group with quadratic Dehn function and undecidable conjugacy problem. This solves Rips’ problem formulated in 1994.

The isoperimeric spectrum consists of all real positive numbers $\alpha$ such that $O(n^\alpha)$ is the Dehn function of a finitely presented group. In this note we show how a recent result of Olshanskii completes the description of the isoperimetric spectrum modulo the celebrated Computer Science conjecture (and one of the seven Millennium Problems) $\mathbf{P=NP}$ and even a formally weaker conjecture., Comment: 3 pages

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., 29 pages, 8 figures

In this paper, we classify all twisted torus knots which are doubly middle Seifert-fibered. Also we show that all of these knots possibly except a few admit Dehn surgery producing a non-Seifert-fibered graph manifold which consists of two Seifert-fibered spaces over the disk with two exceptional fibers, glued together along their boundaries. This provides another infinite family of knots in admitting Dehn surgery yielding such manifolds as done in [5].

We use intersections with horizontal manifolds to show that high-dimensional cycles in the Heisenberg group can be approximated efficiently by simplicial cycles. This lets us calculate all of the higher-order Dehn functions of the Heisenberg groups, thus proving a conjecture of Gromov.

We study a series of 2-generator Sol-manifolds depending on a positive integer n, introduced by Molnar and Szirmai. We construct them as tetrahedron manifolds and show that they are twofold coverings of the 3-sphere branched over specified links. Finally, we give a surgery description of the considered 3-manifolds; indeed, they can be obtained by n−2 and 0 Dehn surgeries along the components of the Whitehead link.

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., Comment: 34 pages, no figure (v2: various corrections and minor improvements especially in the last section)

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., 49 pages, 9 figures, revised version, to appear in Annals of Mathematics

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 borderline case of a result about the sharp isoperimetric constant which implies Gromov hyperbolicity. Our result moreover provides a large scale analog of a recent result of Lytchak and the author which characterizes proper ${\rm CAT}(0)$ in terms of the growth of the Dehn function at all scales. We finally obtain a generalization of this result of Lytchak and the author. Namely, we show that if the Dehn function of a proper, geodesic metric space is sufficiently close to the Euclidean Dehn function up to some scale then the space is not far (in a suitable sense) from being ${\rm CAT}(0)$ up to that scale., Comment: Added Theorems 1.3, 7.1, and 7.2 which provide "bounded-scale" and "coarse" analogs of the previous main theorem. Slightly changed title to reflect the fact that the results also apply to bounded scales

A discrete group with word-length (G, L) is [Formula: see text]-isocohomological for a bounding classes [Formula: see text] if the comparison map from [Formula: see text]-bounded cohomology to ordinary cohomology (with coefficients in ℂ) is an isomorphism; it is strongly [Formula: see text]-isocohomological if the same is true with arbitrary coefficients. In this paper we establish some basic conditions guaranteeing strong [Formula: see text]-isocohomologicality. In particular, we show strong [Formula: see text]-isocohomologicality for an FP∞ group G if all of the weighted G-sensitive Dehn functions are [Formula: see text]-bounded. Such groups include all [Formula: see text]-asynchronously combable groups; moreover, the class of such groups is closed under constructions arising from groups acting on an acyclic complex. We also provide examples where the comparison map fails to be injective, as well as surjective, and give an example of a solvable group with quadratic first Dehn function, but exponential second Dehn function. Finally, a relative theory of [Formula: see text]-bounded cohomology of groups with respect to subgroups is introduced. Relative isocohomologicality is determined in terms of a new notion of relative Dehn functions and a relativeFP∞ property for groups with respect to a collection of subgroups. Applications for computing [Formula: see text]-bounded cohomology of groups are given in the context of relatively hyperbolic groups and developable complexes of groups.

The subgroup of homeoniorphisni group generated by Dehn twists along the set of simple closed pairwise non-homotopic curves with some conditions is studied. It is proved that this group is isomorphic to a free Abelian group of rank k, where k is the number of curves in the set. In the case of an orientable surface, this result is classical.

It is shown that D. Cohen's inequality bounding the isoperimetric function of a group by the double exponential of its isodiametric function is valid in the more general context of locally finite simply connected complexes. It is shown that in this context this bound is ‘best possible’. Also studied are second-dimensional isoperimetric functions for groups and complexes. It is shown that the second-dimensional isoperimetric function of a group is bounded by a recursive function. By a similar argument it is shown that the area distortion of a finitely presented subgroup of a finitely presented group is recursive. Cohen's inequality is extended to second-dimensional isoperimetric and isodiametric functions of 2-connected simplicial complexes.

Stackability for finitely presented groups consists of a dynamical system that iteratively moves paths into a maximal tree in the Cayley graph. Combining with formal language theoretic restrictions yields auto- or algorithmic stackability, which implies solvability of the word problem. In this paper we give two new characterizations of the stackable property for groups, and use these to show that every HNN extension of a stackable group is stackable. We apply this to exhibit a wide range of Dehn functions that are admitted by stackable and autostackable groups, as well as an example of a stackable group with unsolvable word problem. We use similar methods to show that there exist finitely presented metabelian groups that are non-constructible but admit an autostackable structure., 34 pages