Conjugacy problem in groups with quadratic Dehn function

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.

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

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.

Conjugacy problem in groups of oriented geometrizable 3-manifolds

<p>In this paper we investigate the structure of ring diagrams with periodic labels in groups with small cancellation conditions C(3)-T(6). These diagrams are used to solve the power conjugacy search problem in a cyclic subgroup and the problem of power conjugation. In groups of this class the first problem is positively solved. The second is formulated as follows.</p><p>To ascertain whether there are m, n, integers for which the degrees of v, w words with m, n exponents, respectively, are conjugated in the group G = (X; R). To solve this problem in the group with small cancellation conditions C(3)-T(6) it is sufficient to obtain upper bounds for the lengths of the boundary labels of the ring diagram. Therein lies this work.</p><p>Previously, it was proved that for any w word we can take a so-called normal form | word with the following property: each degree of normal form is R-irreducible. Studying of the ring diagrams with irreducible periodic labels manages to break a set of these diagrams into three classes. Working with one of these classes and using the periodicity of the boundary labels of diagram, it is possible to prove the periodicity of the layers of this diagram, and further limit the length of the boundary labels.</p><p>Thus, it turns out that to solve the power conjugacy problem is enough to use the finite known in advance number of checks for the conjugation of the certain degrees of v, w words. In the considered class of groups this problem is solved long ago.</p>

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 introduce a new invariant of bipartite chord diagrams and use it to construct the first examples of groups with Dehn function n2log n. Some of these groups have undecidable conjugacy problem. Our groups are multiple HNN extensions of free groups. We show that n2log n is the smallest Dehn function of a multiple HNN extension of a free group with undecidable conjugacy problem.

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.