Special Issue: International Conference on Group Theory "Combinatorial, Geometric and Dynamical Aspects of Infinite Groups"; Guest Editors: Laurent Bartholdi, Tullio Ceccherini-Silberstein, Tatiana Smirnova-Nagnibeda, Andrzej ŻukNo Access

GEOMETRICAL APPROACH TO THE FREE SOLVABLE GROUPS

A. M. VERSHIK

Mathematical Institute of Russian Academy of Science, St. Petersburg, 191023, Fontanka 27, Russia

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and

S. V. DOBRYNIN

Mathematical Department, St. Petersburg State University, Russia

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https://doi.org/10.1142/S0218196705002657Cited by:4 (Source: Crossref)

Abstract

We give a topological interpretation of the free metabelian group, following the plan described in [11, 12]. Namely, we represent the free metabelian group with d-generators as an extension of the group of the first homology of the d-dimensional lattice (as Cayley graph of the group ℤ^d), with a canonical 2-cocycle. This construction opens a possibility to study metabelian groups from new points of view; in particular to give useful normal forms of the elements of the group, leading to 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 studying the wide class of groups which include the class of all solvable groups.

Keywords:

AMSC: 13D05, 20F16

References

G. P. Egorychev , Amer. Math. Soc. Transl., Math. Monogr. , 2nd edn. 59 ( Providence , 1989 ) . Google Scholar
A. G. Erschler, On asymptotic of drift, Proceedings of POMI Seminars, Vol 283 (St. Petersburg, 2001). pp. 251–257; English translation in J. Math. Sci. (2004) . Google Scholar
D. B. Fuchs and V. A. Rokhlin (eds.) , Topology II: Homotopies and Homologies , Encyclopaedia of Mathematical Sciences 24 ( Springer , 2004 ) . Google Scholar
A. G. Kurosh , Theory of Groups I/II ( Chelsea , 1979 ) . Google Scholar
R. C. Lyndon and P. E. Schupp , Combinatorial Group Theory ( Springer , 1977 ) . Crossref, Google Scholar
A. I. Malcev, Soviet Math. Dolk. 1, 65 (1960). Web of Science, Google Scholar
W. Magnus , A. Karrass and D. Solitar , Combinatorial Group Theory ( Wiley , 1966 ) . Google Scholar
V. M. Petrogradsky, Uspekhi Mat. Nauk 48(5), 181 (1993). Google Scholar
A. L. Shmel'kin, Izv. Akad. Nauk SSSR Ser. Mat. 29(1), 149 (1965). Google Scholar
V. G. Sokolov, Algebra i Logika 8(3), (1969). Google Scholar
A. M. Vershik, Uspekhi Mat. Nauk 55(4,334), 59 (2000). Crossref, Google Scholar
A. M. Vershik , Geometry and dynamics on the free solvable groups , Erwin Schroedinger Institute, No. 899 ( Vienna , 1999 ) . Google Scholar
A. M. Vershik, Selecta Math. Sov. 2(4), 311 (1982). Web of Science, Google Scholar