Geometrical approach to the free sovable groups

Geometrical equivalence of nilpotent torsion free groups

The variety of nilpotent groups is Noetherian. That is why two nilpotent class s groups are geometrically equivalent if and only if they have same quasi-identities ([Pl3]). Therefore, we can describe classes of geometrical equivalence of nilpotent groups instead of describing quasivarieties generated by single nilpotent group. The problem of geometrical equivalence can be researched by the technique of approximation of groups ([Pl2]). In this paper only nilpotent torsion free groups are considered. In the first part two sufficient conditions are presented (Theorem 1 and Theorem 2) when the nil…

The problem of the classification of the nilpotent class 2 torsion free groups up to the geometrically equivalence

In this paper we consider the problem of classification of the nilpotent class 2 finitely generated torsion free groups up to the geometric equivalence. By a very easy technique it is proved that this problem is equivalent to the problem of classification of the complete (in the Maltsev sense) nilpotent torsion free finite rank groups up to the isomorphism. This result, allows us to once more comprehend the complication of the problem of the classification of the quasi-varieties of nilpotent class 2 groups. It is well known that the variety of a nilpotent class s (for every s) groups is Noethe…

Geometric equivalence of $π$-torsion free nilpotent groups

In this paper we study the property of geometric equivalence of groups introduced by B. Plotkin \cite{P1, P2}. Sufficient and necessary conditions are presented for a $π$-torsion-free nilpotent group to be geometrically equivalent to its $π$-completion. We prove that a relatively free nilpotent $π$-torsion-free group and its $π$-completion define the same quasi-variety. Examples of $π$-torsion-free nilpotent groups that are geometrically equivalent to their $π$-completions are given.

The automorphism group of the free group of rank two is a CAT(0) group

We prove that the automorphism group of the braid group on four strands acts faithfully and geometrically on a CAT(0) 2-complex. This implies that the automorphism group of the free group of rank two acts faithfully and geometrically on a CAT(0) 2-complex, in contrast to the situation for rank three and above.

Cleanliness 与 Specialness

Cleanliness versus Specialness

该研究通过证明有限秩自由群的几何洁净图基本群不必为虚紧特殊群，回答了Wise的问题，明确了虚VH-洁净图类是虚几何洁净图类的真子类，推进了群论相关分类研究。

Geometrically infinite surfaces in 3-manifolds with hyperbolic fundamental group

We prove a partial generalization of Bonahon's tameness result to surfaces inside irreducible 3-manifolds with hyperbolic fundamental group. Bonahon's result states that geometrically infinite ends of freely indecomposable hyperbolic 3-manifolds are simply degenerate. It is easy to see that a geometrically infinite end gives rise to a sequence of curves on the corresponding surface whose geodesic representatives are not contained in any compact set. The main step in his proof is showing that one may assume that these curves are simple on the surface. In this paper, we generalize the main step …