On growth of double cosets in hyperbolic groups

Let [Formula: see text] be a hyperbolic group, [Formula: see text] and [Formula: see text] be subgroups of [Formula: see text], and [Formula: see text] be the growth function of the double cosets [Formula: see text]. We prove that the behavior of [Formula: see text] splits into two different cases. If [Formula: see text] and [Formula: see text] are not quasiconvex, we obtain that every growth function of a finitely presented group can appear as [Formula: see text]. We can even take [Formula: see text]. In contrast, for quasiconvex subgroups [Formula: see text] and [Formula: see text] of infinite index, [Formula: see text] is exponential. Moreover, there exists a constant [Formula: see text], such that [Formula: see text] for all big enough [Formula: see text], where [Formula: see text] is the growth function of the group [Formula: see text]. So, we have a clear dichotomy between the quasiconvex and non-quasiconvex case.

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.

Computing with knot quandles

The number [Formula: see text] of colorings of a knot [Formula: see text] by a finite quandle [Formula: see text] has been used in the literature to distinguish between knot types. In this paper, we suggest a refinement [Formula: see text] to this knot invariant involving any computable functor [Formula: see text] from finitely presented groups to finitely generated abelian groups. We are mainly interested in the functor [Formula: see text] that sends each finitely presented group [Formula: see text] to its abelianization [Formula: see text]. We describe algorithms needed for computing the refined invariant and illustrate implementations that have been made available as part of the HAP package for the GAP system for computational algebra. We use these implementations to investigate the performance of the refined invariant on prime knots with [Formula: see text] crossings.

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.

Torsion, torsion length and finitely presented groups

We show that a construction by Aanderaa and Cohen used in their proof of the Higman Embedding Theorem preserves torsion length. We give a new construction showing that every finitely presented group is the quotient of some {C^{\prime}(1/6)} finitely presented group by the subgroup generated by its torsion elements. We use these results to show there is a finitely presented group with infinite torsion length which is {C^{\prime}(1/6)} , and thus word-hyperbolic and virtually torsion-free.

Matricial Representations of Certain Finitely Presented Groups Generated by Order-2 Generators and Their Applications

In this paper, we study matricial representations of certain finitely presented groups Γ2Nwith N-generators of order-2. As an application, we consider a group algebra A2 of Γ22; under our representations. Specifically, we characterize the inverses g-1of all group elements g in Γ22; in terms of matrices in the group algebra A2. From the study of this characterization, we realize there are close relations between the trace of the radial operator of A2; and the Lucas numbers appearing in the Lucas triangle. KEYWORDS: Matricial Representation; Group Presentation; Group Algebras; Lucas Numbers; Lucas Triangle; Finitely Presented Group;Group Relations; Free Probability

Calibrating word problems of groups via the complexity of equivalence relations

(1) There is a finitely presented group with a word problem which is a uniformly effectively inseparable equivalence relation. (2) There is a finitely generated group of computable permutations with a word problem which is a universal co-computably enumerable equivalence relation. (3) Each c.e. truth-table degree contains the word problem of a finitely generated group of computable permutations.

Quasi-isometries between groups with two-ended splittings

We construct a ‘structure invariant’ of a one-ended, finitely presented group that describes the way in which the factors of its JSJ decomposition over two-ended subgroups fit together. For hyperbolic groups satisfying a very general condition, these invariants completely reduce the problem of classifying such groups up to quasi-isometry to a relative quasi-isometry classification of the factors of their JSJ decomposition. Under some additional assumption, our results extend to more general finitely presented groups, yielding a far-reaching generalisation of the quasi-isometry classification of some 3–manifolds obtained by Behrstock and Neumann.The same approach also allows us to obtain such a reduction for the problem of determining when two hyperbolic groups have homeomorphic Gromov boundaries.

Pro-p completions of groups of cohomological dimension 2

We study when an abstract finitely presented group [Formula: see text] of cohomological dimension [Formula: see text] has pro-[Formula: see text] completion [Formula: see text] of cohomological dimension [Formula: see text]. Furthermore, we prove that for a tree hyperbolic limit group [Formula: see text] we have [Formula: see text] and show an example of a hyperbolic limit group [Formula: see text] that is not free and [Formula: see text] is free pro-[Formula: see text]. For a finitely generated residually free group [Formula: see text] that is not a limit group, we show that [Formula: see text] is not free pro-[Formula: see text].