Groups with small Dehn functions and bipartite chord diagrams

For a finite presentation of a group, or more generally, a two-complex, we define a function analogous to the Dehn function that we call the annular Dehn function. This function measures the combinatorial area of maps of annuli into the complex as a function of the lengths of the boundary curves. A finitely presented group has solvable conjugacy problem if and only if its annular Dehn function is recursive.As with standard Dehn functions, the annular Dehn function may change with change of presentation. We prove that the type of function obtained is preserved by change of presentation. Further we obtain upper bounds for the annular Dehn functions of free products and, more generally, amalgamations or HNN extensions over finite subgroups.

Let G be a finitely presented group with solvable word problem. It is of some interest to ask which other decision problems must necessarily be solvable for such a group. Thus it is easy to see that there exist effective procedures to determine whether or not such a group is trivial, or nilpotent of a given class. On the other hand, the conjugacy problem need not be solvable for such a group, for Fridman [5] has shown that the word problem is solvable for the group with unsolvable conjugacy problem given by Novikov [9].

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.

The Lodha–Moore groups provide the first known examples of type [Formula: see text] groups that are non-amenable and contain no non-abelian free subgroups. These groups are related to Thompson’s group [Formula: see text] in certain ways, for instance they contain it as a subgroup in a natural way. We exhibit decompositions of four Lodha–Moore groups, [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], into ascending HNN extensions of isomorphic copies of each other, both in ways reminiscent to such decompositions for [Formula: see text] and also in quite different ways. This allows us to prove two new topological results about the Lodha–Moore groups. First, we prove that they all have trivial homotopy groups at infinity; in particular they are the first examples of groups satisfying all four parts of Geoghegan’s 1979 conjecture about [Formula: see text]. Second, we compute the Bieri–Neumann–Strebel invariant [Formula: see text] for the Lodha–Moore groups, and get some partial results for the Bieri–Neumann–Strebel–Renz invariants [Formula: see text], including a full computation of [Formula: see text].

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.

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 string-rewriting.

Amenability of Schreier graphs and strongly generic algorithms for the conjugacy problem

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

We prove here that a finitely presented group with a free quotient of rank n is an HNN-extension with n stable letters of a finitely generated group where the associated subgroups are finitely generated. This theorem has a number of consequences. In particular, in the event that the free quotient is cyclic it reduces to an elementary and quick proof of a theorem of Bieri and Strebel. 1. Finitely presented groups with free quotients. Our main objective in this note is to prove the following Theorem 1. Let G be a finitely presented group with a free quotient of rank n. Then G is an HNN extension with n stable letters, of a finitely generated group B and finitely generated associated subgroups. It is worth noting that if N is the normal closure in G of B, then N is the fundamental group π(Γ) of a graph Γ of groups, where Γ is the Cayley graph of a free group of rank n. The vertex groups are isomorphic to B and the edge groups are isomorphic to the associated subgroups involved in the decomposition of G as an HNN–extension and are therefore all finitely generated. There are a number of consequences of Theorem 1. The first of these is the following theorem of Bieri and Strebel [3] which is a special case of Theorem 1. Corollary 1. A finitely presented group with an infinite cyclic quotient is an HNN extension of a finitely generated group with a single stable letter and finitely generated associated subgroups. In order to formulate the second consequence of Theorem 1, we recall that a group is termed coherent if all of its finitely generated subgroups Date: November 26, 2006. 1991 Mathematics Subject Classification. Primary 20.

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.

We give a construction of two-sided invariant metrics on free products (possibly with amalgamation) of groups with two-sided invariant metrics and, under certain conditions, on HNN extensions of such groups. Our approach is similar to the Graev’s construction of metrics on free groups over pointed metric spaces.

Let N 1 (N 2) be the normal closure of a finite symmetrized set R 1 (R 2, respectively) in a finitely generated free group F = F(A). As is known, if R i satisfies condition C(6), then the conjugacy problem is decidable in F/N i . In the paper, it is proved that, if one adds to condition C(6) on the set R 1 ∪ R 2 the atoricity condition for the presentation 〈A | R 1, R 2〉, then the conjugacy problem is decidable in the group F/N 1 ∩ N 2 as well. In particular, for the decidability of the conjugacy problem in F/N 1 ∩ N 2, it is sufficient to assume that the set R 1 ∪ R 2 satisfies condition C(7).

In this work we study the results of M. Kassabov and T. Riley in [7], where they prove that Baumslag group has an exponential Dehn function, in contrast to its analog when a torsion relation is added, which satisfies a polynomial isoperimetric inequality.We present some basic concepts of group theory and modules.Additionally we study the contruction of free groups and the concept of presentation of a group in detail.

The theory of groups acting on trees due to Bass and Serre (1969) is applied to simplify some results of Burns (1972, 1973) giving conditions under which an amalgamated free product or HNN extension has the properties that any finitely generated subgroup containing an infinite subnormal subgroup must have finite index and that the intersection of two finitely generated subgroups is finitely generated.

In 1911 M. Dehn posed the word and conjugacy problems for groups in general and provided algorithms which solved these problems for the fundamental groups of closed orientable two-dimensional manifolds. A crucial feature of these groups is that (with trivial exceptions) they are defined by a single relator r with the property that if s is a cyclic conjugate of r or r-1, with s ≠ r-1, there is very little cancellation in forming the product rs. Dehn’s algorithms have been extended to large classes of groups possessing presentations in which the defining relations have a similar small cancellation property. At first, investigations were concerned with the solution of the word problem for groups G presented as small cancellation quotients of a free group F. The theory was subsequently extended to the case where F is a free product, a free product with amalgamation, or an HNN extension. Moreover, strong results were obtained about algebraic properties; for example, one can classify torsion elements and commuting elements in small cancellation quotients.