We give examples of CAT(0), biautomatic, free-by-cyclic, one-relator groups which have finite-rank free subgroups of huge (Ackermannian) distortion. This leads to elementary examples of groups whose Dehn functions are similarly extravagant. This behaviour originates in manifestations of Hercules-versus-the-hydra battles in stringrewriting.

Abstract. The dead-end depth of an element g of a group G, with respect to a generating set A, is the distance from g to the complement of the radius d A (1, g) closed ball, in the word metric d A defined with respect to A. We exhibit a finitely presented group G with a finite generating set with respect to which there is no upper bound on the dead-end depth of elements.

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We prove that every finitely generated nilpotent group of class c admits a polynomial isoperimetric function of degree c + 1 and a linear upper bound on its filling length function.

A diagram is a finite planar 2-complex homeomorphic to a singular 2-disc. A van Kampen diagram is a labelled diagram amounting to a graphical demonstration of how a word in a finite presentation P of a group Γ that represents the identity, is a consequence of the defining relations. (See Definitions 2.1 and 4.1.)The use of diagrams to probe the geometry of finitely presented groups and to produce invariants (filling functions) is well established in Geometric Group Theory. To date, attention has focused on the area of diagrams, and the resulting Dehn function (also known as the minimal isoperimetric function), and on the diameter of the 1-skeleton of diagrams (in the combinatorial metric), which gives the minimal isodiametric function. And recently the filling length of a diagram D has proved important [9, 10]; we define FL(D) to be the minimal length L such that there is a combinatorial null-homotopy of ∂D, across D and down to a base vertex, through loops of length at most L. (See § 2.1 for more details.)The innovation in this article is to bring duality considerations to bear on the study of diagrams. We define the gallery length GL(D) of a diagram D to be the diameter of the 1-skeleton of its dual. We will also define DGL(D) which combines diameter and gallery length as realisable on complementary pairs of maximal trees; see § 2.3. In § 2 we give formal definitions of diagrams and of GL(D) and DGL(D) as well as other diagram measurements, and in Proposition 2.4 we list some of the inequalities that relate diagram measurements. Then in § 3 we use DGL to control the filling length of a diagram.Theorem 3.5. Fix λ > 0. There exists K > 0 such that for every diagram D in which the boundary of each 2-cell has at most λ edges, FL(D) K(DGL(D) + Perimeter(D)).This inequality is an amendment of a suggestion of Gromov [13, § 5.C]: the term DGL(D) replaces Diam(D). Gromov's inequality was known to fail [6] in the context of 2-discs with Riemannian metrics. In § 3 we give our own family of diagrams D n that have bounded perimeter and diameter, and at most three edges in the boundary of each 2-cell, but have filling length tending to infinity. It follows that Gromov's inequality does not hold in a combinatorial context. Our proof that FL(D n ) → ∞ takes us into unlikely territory. We use the following result, proved in an appendix, concerning the number n of terms required to express an integer n as a sum of Mathematics Subject Classification 20F05 (primary), 20F06, 57M05, 57M20 (secondary). s. m. gersten and t. r. riley which GL Q (D w ) = GL Q (w). The relations in Q that involve t are tu −1 and [t ±1 , z]. It follows that all occurrences of t in D w must be in 't-corridors', that is, concatenations of 2-cells with boundary labels [t, z] ±1 and with adjacent 2-cells joined across an edge labelled t. There are two types of t-corridor: linear t-corridors start with a 2-cell labelled by (tu −1 ) ±1 and finish with a 2-cell labelled (tu −1 ) ±1

We construct a hyperbolic group with a hyperbolic subgroup for which inclusion does not induce a continuous map of the boundaries.

We give examples of hyperbolic groups with finite-rank free subgroups of huge (Ackermannian) distortion.

We pose some graph theoretic conjectures about duality and the diameter of maximal trees in planar graphs, and we give innovations in the following two topics in geometric group theory, where the conjectures have applications.Central extensions. We describe an electrostatic model concerning how van Kampen diagrams change when one takes a central extension of a group. Modulo the conjectures, this leads to a new proof that finitely generated class c nilpotent groups admit degree c + 1 polynomial isoperimetric functions.Filling functions. We collate and extend results about interrelationships between filling functions for finite presentations of groups. We use the electrostatic model in proving that the gallery length filling function, which measures the diameter of the duals of diagrams, is qualitatively the same as a filling function DlogA, concerning the sum of the diameter with the logarithm of the area of a diagram. We show that the conjectures imply that the space-complexity filling function filling length essentially equates to gallery length. We give linear upper bounds on these functions for a number of classes of groups including fundamental groups of compact geometrizable 3-manifolds, certain graphs of groups, and almost convex groups. Also we define restricted filling functions which concern diagrams with uniformly bounded vertex valence, and we show that, assuming the conjectures, they reduce to just two filling functions-the analogues of non-deterministic space and time.