arXiv subjects
Publications and source records attributed to Olga Kulikova.
This paper describes some generalizations of the results presented in the book "Geometry of defining Relations in Groups" , of A.Yu.Ol'shanskii to the case of non-cyclic torsion-free hyperbolic groups. In particular, it is proved that for every non-cyclic torsion-free hyperbolic group, there exists a non-Abelian torsion-free quotient group in which all proper subgroups are cyclic, and the intersection of any two of them is not trivial.
Consider a presentation $\mathcal{P}=<{\bf x}\mid{\bf \bigcup_{i=1}^n r_i}>$. Let ${\bf R_i}$ be the normal closure of the set ${\bf r_i}$ in the free group ${\bf F}$ with basis ${\bf x}$, $\mathcal{P}_i=<{\bf x}\mid{\bf r_i}>$, ${\bf N_i} = \prod_{j\neq i}{\bf R_j}$. In the present article, using geometric techniques of pictures, generators for $\frac{{\bf R_i}\cap {\bf N_i}}{[{\bf R_i}, {\bf N_i}]}$, $i=1,...,n$, are obtained from a set of generators over $\{\mathcal{P}_i\mid i=1,..., n\}$ for $π_2(\mathcal{P})$. As a corollary, we get a sufficient condition for the family $\{{\bf R_1},...,{\bf R_n}\}$ to be independent.