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Steklov Institute of Mathematics at St.Petersburg

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PREPRINT 14/2004

A. M. Vershik and S. V. Dobrynin
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TOPOLOGICAL APPROACH TO THE FREE SOLVABLE GROUPS

This preprint was accepted September 20, 2004

ABSTRACT:
We give a topological interpretation of the free metabelian group, following the plan
described in \cite{Ver1},\cite{Ver2}. Namely we represent the free metabelian group with
$d$-generators as extension the group of the first homology of the $d$-dimensional lattice as
Cayley graph of the group ${\Bbb Z}^d$ with a canonical 2-cocycle. This construction open the
possibility to study metabelian groups from new points of view; in particular to give a useful
normal forms of the elements of the group, applications to the random walks and so on. We also
describe the satellite groups which correspond to all 2-cocycles of cohomology group associated
with the free solvable groups. The homology of the Cayley graph can be used for the studying of the
wide class of groups which including the class of all solvable groups.
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