\titleofpresentation{Quotients of large Lie Groups and Lie Algebras}

\presenter{A. Yu. Olshanskii}

We present recent results obtained jointly with D. I. Osin and 
Yu. A. Bahturin.

Recall that a group $G$ is {\it large} if some finite index subgroup
of $G$ admits a surjective homomorphism onto a non-cyclic free
group. Given a subset $S$ of a group $G$, we denote by $\langle 
S\langle ^G$ the
normal closure of $S$ in $G$. In the recent paper \cite{OlshanskiiL},
Lackenby proved the following theorem.


Let $G$ be a large group, $H$ is a normal subgroup of $G$ of
finite index admitting a surjective homomorphism onto a
non-cyclic free group, $g_1, \ldots , g_k$ a collection of
elements of $H$. Then the quotient group $G/\langle g_1^{n}, \dots ,
g_k^{n}\rangle^G$ is large for all but finitely many $n\in \mathbb
N$.

We provide a shorter proof of this theorem based just on the
Schreier formula and Baumslag--Pride theorem of 1978 on the
largeness of a group given by $n\ge 2$ generators and $n-2$
relations.

Thus our approach is very elementary. However it leads to new
constructions. We obtain new examples of finitely generated
infinite torsion groups. In particular, our examples enjoys
in addition the following properties: the orders of elements are square free;
every nilpotent section (i.e., a factor-group of a subgroup)
is abelian; every section is residually finite. 

Also an analog of the theorem and other results are obtained for restricted
Lie algebras over fields of prime characteristic.

\begin{ourbiblio}{9}%{ECHPT}
%\label{bibbb}

\bibitem{OlshanskiiL} M. Lackenby. Adding high powered relations to 
large 
groups. Prep., 2005; arXiv:
mathGR/0512356.

\bibitem{OlshanskiiO} A. Yu. Olshanskii and D. V. Osin. Large groups and 
their periodic quotients. To appear in Proc. of the AMS.
7 pp. 

\bibitem{BOlshanskii}Yu. A. Bahturin and A. Yu. Olshanskii. Large 
restricted 
Lie algebras. To appear in Journal of Algebra, 19 pp.

\end{ourbiblio}

\presenteraddress{Vanderbilt University and Moscow State University}

\email{alexander.olshanskiy@vanderbilt.edu}

