We construct the first examples of finitely presented groups with quadratic Dehn function containing a finitely generated infinite torsion subgroup. These examples are “optimal” in the sense that the Dehn function of any such finitely presented group must be at least quadratic. Moreover, we show that for any n ≥ 2 48 n\geq 2^{48} such that n n is either odd or divisible by 2 9 2^9 , any infinite free Burnside group with exponent n n is a quasi-isometrically embedded subgroup of a finitely presented group with quadratic Dehn function satisfying the Congruence Extension Property.

Abstract We prove that Thompson’s group T and, more generally, all the Higman–Thompson groups $T_n$ have quadratic Dehn function.

Finitely Presented Simple Groups with at Least Exponential Dehn Function

A semigroup with linearithmic Dehn function

On the Dehn functions of a class of monadic one-relation monoids

Abstract The algebraic mapping torus $M_{\Phi }$ of a group $G$ with an automorphism $\Phi$ is the HNN-extension of $G$ in which conjugation by the stable letter performs $\Phi$ . We classify the Dehn functions of $M_{\Phi }$ in terms of $\Phi$ for a number of right-angled Artin groups (RAAGs) $G$ , including all $3$ -generator RAAGs and $F_k \times F_l$ for all $k,l \geq 2$ .

The group \mathfrak{X}(G) is obtained from G\ast G by forcing each element g in the first free factor to commute with the copy of g in the second free factor. We make significant additions to the list of properties that the functor \mathfrak{X} is known to preserve. We also investigate the geometry and complexity of the word problem for \mathfrak{X}(G) . Subtle features of \mathfrak{X}(G) are encoded in a normal abelian subgroup W<\mathfrak{X}(G) that is a module over \mathbb{Z} Q , where Q= H_1(G,\mathbb{Z}) . We establish a structural result for this module and illustrate its utility by proving that \mathfrak{X} preserves virtual nilpotence, the Engel condition, and growth type – polynomial, exponential, or intermediate. We also use it to establish isoperimetric inequalities for \mathfrak{X}(G) when G lies in a class that includes Thompson's group F and all non-fibred Kähler groups. The word problem is soluble in \mathfrak{X}(G) if and only if it is soluble in G . The Dehn function of \mathfrak{X}(G) is bounded below by a cubic polynomial if G maps onto a non-abelian free group.

Abstract Consider the following classes of pairs consisting of a group and a finite collection of subgroups:• $ \mathcal{C}= \left \{ (G,\mathcal{H}) \mid \text{$\mathcal{H}$ is hyperbolically embedded in $G$} \right \}$ • $ \mathcal{D}= \left \{ (G,\mathcal{H}) \mid \text{the relative Dehn function of $(G,\mathcal{H})$ is well-defined} \right \} .$ Let $G$ be a group that splits as a finite graph of groups such that each vertex group $G_v$ is assigned a finite collection of subgroups $\mathcal{H}_v$ , and each edge group $G_e$ is conjugate to a subgroup of some $H\in \mathcal{H}_v$ if $e$ is adjacent to $v$ . Then there is a finite collection of subgroups $\mathcal{H}$ of $G$ such that1.If each $(G_v, \mathcal{H}_v)$ is in $\mathcal C$ , then $(G,\mathcal{H})$ is in $\mathcal C$ .2.If each $(G_v, \mathcal{H}_v)$ is in $\mathcal D$ , then $(G,\mathcal{H})$ is in $\mathcal D$ .3.For any vertex $v$ and for any $g\in G_v$ , the element $g$ is conjugate to an element in some $Q\in \mathcal{H}_v$ if and only if $g$ is conjugate to an element in some $H\in \mathcal{H}$ .That edge groups are not assumed to be finitely generated and that they do not necessarily belong to a peripheral collection of subgroups of an adjacent vertex are the main differences between this work and previous results in the literature. The method of proof provides lower and upper bounds of the relative Dehn functions in terms of the relative Dehn functions of the vertex groups. These bounds generalize and improve analogous results in the literature.

For a finitely generated group G and collection of subgroups \mathcal{P} , we prove that the relative Dehn function of a pair (G,\mathcal{P}) is invariant under quasi-isometry of pairs. Along the way, we show quasi-isometries of pairs preserve almost malnormality of the collection and fineness of the associated coned-off Cayley graphs. We also prove that for a cocompact simply connected combinatorial G - 2 -complex X with finite edge stabilisers, the combinatorial Dehn function is well defined if and only if the 1 -skeleton of X is fine. We also show that if H is a hyperbolically embedded subgroup of a finitely presented group G , then the relative Dehn function of the pair (G, H) is well defined. In the appendix, it is shown that the Baumslag–Solitar group \mathrm{BS}(k,l) has a well-defined Dehn function with respect to the cyclic subgroup generated by the stable letter if and only if neither k divides l nor l divides k .

For every k ⩾ 3 $k\geqslant 3$ , we exhibit a simply connected k $k$ -nilpotent Lie group N k $N_k$ whose Dehn function behaves like n k $n^k$ , while the Dehn function of its associated Carnot graded group gr ( N k ) $\mathsf {gr}(N_k)$ behaves like n k + 1 $n^{k+1}$ . This property and its consequences allow us to reveal three new phenomena. First, since those groups have uniform lattices, this provides the first examples of pairs of finitely presented groups with bi-Lipschitz asymptotic cones but with different Dehn functions. The second surprising feature of these groups is that for every even integer k ⩾ 4 $k \geqslant 4$ , the centralised Dehn function of N k $N_k$ behaves like n k − 1 $n^{k-1}$ and so has a different exponent than the Dehn function. This answers a question of Young. Finally, we turn our attention to sublinear bi-Lipschitz equivalences (SBEs). Introduced by Cornulier, these are maps between metric spaces inducing bi-Lipschitz homeomorphisms between their asymptotic cones. These can be seen as weakenings of quasi-isometries where the additive error is replaced by a sublinearly growing function v $v$ . We show that a v $v$ -SBE between N k $N_k$ and gr ( N k ) $\mathsf {gr}(N_k)$ must satisfy v ( n ) ≽ n 1 / ( 2 k + 2 ) $v(n)\succcurlyeq n^{1/(2k + 2)}$ , strengthening the fact that those two groups are not quasi-isometric. This is the first instance where an explicit lower bound is provided for a pair of SBE groups.

We construct and study finitely presented groups with quadratic Dehn function (QD-groups) and present the following applications of the method developed in our recent papers. (1) The isomorphism problem is undecidable in the class of QD-groups. (2) For every recursive function f , there is a QD-group G containing a finitely presented subgroup H whose Dehn function grows faster than f . (3) There exists a group with undecidable conjugacy problem but decidable power conjugacy problem; this group is QD.

We develop new methods for computing the precise Dehn functions of coabelian subgroups of direct products of groups, that is, subgroups which arise as kernels of homomorphisms from the direct product onto a free abelian group. These improve and generalise previous results by Carter and Forester on Dehn functions of level sets in products of simply connected cube complexes, by Bridson on Dehn functions of cocyclic groups and by Dison on Dehn functions of coabelian groups. We then provide several applications of our methods to subgroups of direct products of free groups, to groups with interesting geometric finiteness properties and to subgroups of direct products of right-angled Artin groups.

We demonstrate under appropriate finiteness conditions that a coarse embedding induces an inequality of homological Dehn functions. Applications of the main results include a characterization of what finitely presentable groups may admit a coarse embedding into a hyperbolic group of geometric dimension 2, characterizations of finitely presentable subgroups of groups with quadratic Dehn function with geometric dimension 2, and to coarse embeddings of nilpotent groups into other nilpotent groups of the same growth and into hyperbolic groups.

In contrast to being automatic, being Cayley automatic a priori has no geometric consequences. Specifically, Cayley graphs of automatic groups enjoy a fellow traveler property. Here, we study a distance function introduced by the first author and Trakuldit which aims to measure how far a Cayley automatic group is from being automatic, in terms of how badly the Cayley graph fails the fellow traveler property. The first author and Trakuldit showed that if it fails by at most a constant amount, then the group is in fact automatic. In this paper, we show that for a large class of non-automatic Cayley automatic groups this function is bounded below by a linear function in a precise sense defined herein. In fact, for all Cayley automatic groups which have super-quadratic Dehn function, or which are not finitely presented, we can construct a non-decreasing function which (1) depends only on the group and (2) bounds from below the distance function for any Cayley automatic structure on the group.

Abstract 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.

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 that of the associated symmetric space, and when n is equal to the rank, we show that the n-dimensional filling volume function grows exponentially. This broadly generalizes a theorem of Lubotzky-Mozes-Raghunathan on length distortion in lattices and confirms conjectures of Thurston, Gromov, and Bux-Wortman.

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 $$\Gamma $$ .

We present a metric condition $\TTMetric$ which describes the geometry of classical small cancellation groups and applies also to other known classes of groups such as two-dimensional Artin groups. We prove that presentations satisfying condition $\TTMetric$ are diagrammatically reducible in the sense of Sieradski and Gersten. In particular, we deduce that the standard presentation of an Artin group is aspherical if and only if it is diagrammatically reducible. We show that, under some extra hypotheses, $\TTMetric$-groups have quadratic Dehn functions and solvable conjugacy problem. In the spirit of Greendlinger's lemma, we prove that if a presentation P = 〈X| R〉 of group G satisfies conditions $\TTMetric -C'(\frac {1}{2})$, the length of any nontrivial word in the free group generated by X representing the trivial element in G is at least that of the shortest relator. We also introduce a strict metric condition $\TTMetricStrict$, which implies hyperbolicity.

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 a 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 particular braid groups and spherical Artin groups) and weak Garside groups, including e.g. fundamental groups of the complements of complexified finite simplicial arrangements of hyperplanes, braid groups of well-generated complex reflection groups, and one-relator groups with non-trivial center. Among the results are: biautomaticity, existence of EZ and Tits boundaries, the Farrell–Jones conjecture, the coarse Baum–Connes conjecture, and a description of higher order homological and homotopical Dehn functions. As a means of proving the Helly property we introduce and use the notion of a (generalized) cell Helly complex.

Polynomial-time proofs that groups are hyperbolic