Narrow Results

We examine a certain embedding problem for groups that have a presentation described by partial automorphisms. Semidirect products and HNN extensions have such a presentation. The embedding problem is closely related to systems of partial automorphisms, which are formalized by the concept of inverse semigroups. A necessary and sufficient condition for a group to be embedded in a certain sense is obtained by geometric methods using van Kampen diagrams.

Lyndon's group F^Z[x] is the free exponential group over the ring of integral polynomials Z[x]. This group, introduced by Lyndon in the 1960s, continues to be of interest to group theorists due to its importance in the study of first-order properties of free groups, in particular, equations over free groups. One of the crucial results of Lyndon's study was that the group F^Z[x] is fully residually F; i.e. for any finite collection of nontrivial elements in F^Z[x] there exists a homomorphism φ : F^Z[x] → F which is the identity on F and maps the given elements of F^Z[x] into nontrivial elements of F. The importance of F^Z[x] was further emphasized when Kharlampovich and Myasnikov proved in [3] that a finitely generated group is fully residually free if and only if it is embeddable into F^Z[x]. Lyndon's group and its subgroups play a vital role in the technique employed by O. Kharlampovich and A. Myasnikov in their solution of the famous Tarski problem on the decidability of the elementary theory of a free group (see [4, 5]).

In this paper, we show that Lyndon's group is conjugately residually free, i.e. it is possible to map F^Z[x] to the free group F preserving the nonconjugacy of two elements. This result is a further step towards the understanding of the properties of F^Z[x]; moreover, it is closely related to the problem of "lifting solutions" of equations from F to F^Z[x], since our result implies that the solutions can indeed be "lifted" from F to F^Z[x] for equations of the type x^-1 c₁ x = c₂.

The structure of Lyndon's group, described by A. Myasnikov and V. Remeslennikov in [8], involves an infinite sequence of free constructions of a specific type, called free extensions of centralizers. For more results on residual properties of certain types of free constructions, see also the works of Ribes, Segal and Zalesskii (for example [9]).

We work in the density model of random groups. We prove that they satisfy an isoperimetric inequality with sharp constant 1-2d depending upon the density parameter d. This implies in particular a property generalizing the ordinary C′ small cancellation condition, which could be termed "macroscopic cancellation". This also sharpens the evaluation of the hyperbolicity constant δ.

As a consequence we get that the standard presentation of a random group at density d < 1/5 satisfies the Dehn algorithm and Greendlinger's lemma, and that it does not for d > 1/5.

For this we establish a version of the local-global principle for hyperbolic spaces (Cartan–Hadamard–Gromov theorem) involving arbitrarily small loss in the isoperimetric constant.

We define the notion of V(6) presentations that naturally generalize C(4)&T(4) and C(6) small-cancellation presentations. We consider V(6) groups (i.e. groups having V(6) presentations) where every piece is of length one. No natural geometrical interpretation is known for this class of groups. However, by using combinatorial and diagrammatical arguments, we construct a biautomatic structure for these groups.

It is conjectured that an irreducible Artin group which is of infinite type has trivial center. The conjecture is known to be true for two-dimensional Artin groups and for a few other types of Artin groups. In this work, we show that the conjecture holds true for Artin groups which satisfy a condition stronger than being of infinite type. We use small cancellation theory of relative presentations.

In this work we define Magnus subsemigroups of one-relator groups as a generalisation of Magnus subgroups. In contrast with Magnus subgroups which by the Dehn-Magnus Freiheitssatz are freely generated, Magnus subsemigroups are not necessarily free. We give a necessary and sufficient condition for a Magnus subsemigroup to be free in terms of the combinatorial structure of the defining relator R, provided that the given presentation satisfies the small cancellation condition C(6)&T(4) and R is not exceptional. We use word combinatorics with analysis of van Kampen diagrams, as developed in [1].

We show that the twisted conjugacy problem for extra-large Artin groups and for the class of CLTTF Artin groups introduced by J. Crisp in Ref. 18 is solvable, relative to length preserving automorphisms. We also show that these groups have infinite number of such twisted conjugacy classes and twisted conjugacy of elements of a standard parabolic subgroup which is invariant under the automorphism in the group, implies twisted conjugacy already in the parabolic subgroup.