Dehn functions of finitely presented metabelian groups

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 conjugacy problem for Higman’s group

Higman’s group [Formula: see text] is a remarkable group with large (non-elementary) Dehn function. Higman constructed the group in 1951 to produce the first examples of infinite simple groups. Using finite state automata, and studying fixed points of certain finite state transducers, we show the conjugacy problem in [Formula: see text] is decidable for all inputs. Diekert, Laun and Ushakov have recently shown the word problem in [Formula: see text] is solvable in polynomial time, using the power circuit technology of Myasnikov, Ushakov and Won. Building on this work, we also show in a strongly generic setting that the conjugacy problem for [Formula: see text] has a polynomial time solution.

Dehn functions and Hölder extensions in asymptotic cones

The Dehn function measures the area of minimal discs that fill closed curves in a space; it is an important invariant in analysis, geometry, and geometric group theory. There are several equivalent ways to define the Dehn function, varying according to the type of disc used. In this paper, we introduce a new definition of the Dehn function and use it to prove several theorems. First, we generalize the quasi-isometry invariance of the Dehn function to a broad class of spaces. Second, we prove Hölder extension properties for spaces with quadratic Dehn function and their asymptotic cones. Finally, we show that ultralimits and asymptotic cones of spaces with quadratic Dehn function also have quadratic Dehn function. The proofs of our results rely on recent existence and regularity results for area-minimizing Sobolev mappings in metric spaces.

Conjugacy problem in groups with quadratic Dehn function

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

Taming the hydra: The word problem and extreme integer compression

For a finitely presented group, the word problem asks for an algorithm which declares whether or not words on the generators represent the identity. The Dehn function is a complexity measure of a direct attack on the word problem by applying the defining relations. Dison and Riley showed that a “hydra phenomenon” gives rise to novel groups with extremely fast growing (Ackermannian) Dehn functions. Here, we show that nevertheless, there are efficient (polynomial time) solutions to the word problems of these groups. Our main innovation is a means of computing efficiently with enormous integers which are represented in compressed forms by strings of Ackermann functions.

Dehn Functions

This chapter is concerned with Dehn functions. It begins by presenting jigsaw puzzles that are somewhat different from the conventional kind and explains how to solve them. It then considers a complexity measure for the word problem and shows that, for a word w, the problem of finding a sequence of free reductions, free expansions, and applications of defining relators that carries it to the empty word is equivalent to solving the puzzle where, starting from some vertex υ‎, one reads w around the initial circle of rods. The chapter also explains how the Dehn function corresponds to an isoperimetric problem in a combinatorial space, the Cayley 2-complex, and describes a continuous version of this, via group actions, along with the isoperimetry in Riemannian manifolds. Finally, it defines the Dehn function as a quasi-isometry invariant. The discussion includes exercises and research projects.

A Dehn function for Sp(2n, ℤ)

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.

The large-scale geometry of locally compact solvable groups

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.