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