Problems of growth and rationality in algebra and topology. (English. Russian original) Zbl 0607.55007

Babenko, I. K.

Russ. Math. Surv. 41, No. 2, 117-175 (1986); translation from Usp. Mat. Nauk 41, No. 2(248), 95-142 (1986).

This paper is a complete and systematic account in the range of problems of rationality and growth in algebra and topology. Indeed, various problems of algebra and topology lead to the study of sequences of integers closely connected with the initial object. For instance, if X is a finite complex the analytic properties of Poincare series, \(\sum^{\infty}_{i=1}\dim H_ i(\Omega X,k) t^ i\), led over the last thirty years to a number of conjectures and problems. For finitely generated groups, J. Milnor stated, in 1968, the problem of the existence of intermediate growth. The author gives the reader precise definitions, a list of these problems, their interdependence and describes some examples. A bibliography of 128 titles ends the text.

Reviewer: J.C.Thomas

MSC:

55P62	Rational homotopy theory
16E40	(Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
57M05	Fundamental group, presentations, free differential calculus
20F99	Special aspects of infinite or finite groups

Keywords:

Hilbert series; growth of group; finite complex; Poincare series