Arye Juhász

Solution of the Conjugacy Problem in One-Relator Groups

In this talk I sketch a proof of the following theorem: Theorem. One-relator groups have solvable conjugacy problems.

Nonsingular Equations over Groups II

Let G be a group, t an element distinct from G, and r(t) = g 1 t l 1 …g k t l k ∈ G* ⟨t⟩, where each g i is an element of G order greater than 2, and the l i are nonzero integers such that l 1 + l 2+…+l k ≠ 0 and |l i | ≠ |l j | for i ≠ j. We prove that if k = 4, then the natural map from G to the one-relator product ⟨G*t | r(t)⟩ is injective. This together with previous results show that the natural map from G is injective for k ≥ 1.

Non-singular Equations over Groups I

Let G be a group, t an element distinct from G and r(t)= g1tl1 ⋯ gktlk∈ G ∗ 〈t 〉, where each gi is an element of G of order greater than 2 and the li are non-zero integers such that l1+l2+ ⋯ +lk≠ 0 and |li| ≠ |lj| for i ≠ j. It is known that if k≤ 2, then the natural map from G to the one-relator product 〈G,t | r(t)〉 is injective. In this paper, we prove that the same holds for all k ∉ {4, 5}.

ON A FREIHEITSSATZ FOR CYCLIC PRESENTATIONS

In this paper we examine the embeddability of a certain one-relator group in a cyclically presented group, which is naturally associated to it, as a function of a certain subgroup of the one-relator group. We find that if this subgroup is largest possible then it is rare that embedding occurs. On the other hand, when this subgroup is smallest possible, then embedding is very likely.

SOME REMARKS ON ONE-RELATOR AMALGAMATED FREE PRODUCTS II

We consider quotients of the free product G of groups A and B, amalgamated along a group C, which is malnormal in A and in B. We concentrate on quotients H of G by the normal closure of a single relator which is a power of a word. We show that a part of the results which are true for the corresponding one relator free products hold true for H. We reduce the problems to the free product case and use small cancellation theory.

The infinite Fibonacci groups and relative asphericity

We prove that the generalised Fibonacci group F (r, n) is infinite for (r, n) ∈ {(7 + 5k, 5), (8 + 5k, 5) : k ≥ 0}. This together with previously known results yields a complete classification of the finite F (r, n), a problem that has its origins in a question by J H Conway in 1965. The method is to show that a related relative presentation is aspherical from which it can be deduced that the groups are infinite.

EXTENSION OF GROUP PRESENTATIONS AND RELATIVE SMALL CANCELLATION THEORY I

In this paper we introduce extensions of group presentations with a corresponding small cancellation theory and show how this can be used in order to reduce geometrical and combinatorial problems in certain groups, to the same problems in simpler groups. Let us illustrate this by the following simple example: let P1 = 〈a, b, c, d|R〉 be a presentation of G1, where R = (ab 2c−1dc)7(bd−1c3b)6 and let P2 = 〈x, y|xy〉 be a presentation of G2. We show that since R = A B where A = ab2c−1dc and B = bd−1c3b and since the map φ : F2 → F1 defined by φ(x) = A and φ(y) = B is a monomorphism satisfying R ∈ φ(N2) ⊆ N1, where F2 = 〈x, y|−〉, F1 = 〈a, b, c, d|−〉, N1 is the normal closure of R in F1 and N2 is the normal closure of x y in F2, it follows that G1 is biautomatic, because G2 is biautomatic. Moreover, for example the subgroup H of G1 generated by C and D where C = abdc −2b3 and D = dc−1a5b−7c is a free subgroup of G1 and the natural map φ̂ : G2 → G1 is an embedding. In the above situation we say that P1 is an extension of P2 via φ. (See 1.1 for the precise definition.) Based on the general idea of derived diagrams due to E. Rips, we develop a relative version of small cancellation presentations which is roughly as follows: assume P1 is an extension of P2. Consider φ(N2) as the set of defining relations for G1 and describe the elements of N1 as products of conjugates of elements from φ(N2). Choose a shortest such product. Then P1 is small cancellation relative to P2 if the cancellation between two factors in a shortest product is small. The main results of this work are collected in Theorem 1.1. Its proof is divided into 2 parts. First we prove that if P1 is a small cancellation presentation relative to P2 then all the results of the Main Theorem hold true (Theorem 2.1). Then, we show that under the conditions of the theorem, P1 is small cancellation relative to P2 (Theorem 1.2).

A Freiheitssatz for Whitehead Graphs of One-Relator Groups with Small Cancellation

In this work we study one-relator groups with a certain small cancellation condition. We focus on the following two general problems: the free subgroups of these groups, and what can be said on the automorphism group of these groups. Both problems are widely open. We introduce a graph-theoretical test which, if successful, implies that the subgroup under consideration is free. We also extend a result due to V. Shpilrain on automorphism groups of one-relator groups.

On the Solvability of Equations Over Groups

Abstract Let G be a group. An equation over G is an expression r(t) = 1 where and ε i ∈ {1, −1} for i = 1,…, n. r(t) = 1 is singular if ε1 + ε2 + · + ε n = 0. Singular equations may have no solutions; e.g., a 1 t −1 a 2 t = 1, where a 1 and a 2 have different orders. In this work we formulate a conjecture which, if true, provides a sufficient condition for the solvability of singular equations in which some of the a i have finite order. We prove that the conjecture holds true for a class of groups which includes abelian and certain locally nilpotent groups.

On the solvability of a class of equations over groups

Let