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On the group of substitutions of formal power series with integer coefficients. (English. Russian original) Zbl 1152.20026

Izv. Math. 72, No. 2, 241-264 (2008); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 72, No. 2, 39-64 (2008).
Let \(k\) be a ring with an identity element and \(k[\![x]\!]\) the ring of formal power series in \(x\) over \(k\). \(\mathcal J(k)\) is defined to be the group of normalised formal power series in \(k[\![x]\!]\) under substitution. That is, the elements of \(\mathcal J(k)\) are formal power series of the form \(f(x)=x+\alpha_1 x^2+\alpha_2x^3+\cdots\). When \(p\) is a prime and \(k=\mathbb{F}_p\) is the finite field of \(p\) elements, \(\mathcal J(\mathbb{F}_p)\) is well studied and known as the ‘Nottingham group’ [see R. Camina in New horizons in pro-\(p\) groups, Prog. Math. 184, 205-221 (2000; Zbl 0977.20020) for a survey of results].
In this paper the authors focus on the case when \(k=\mathbb{Z}\), the integers. The authors begin by proving that as a topological group \(\mathcal J(\mathbb{Z})\) is 4-generator. These generators are not defined unambiguously, and the authors go on to show that for almost every choice of generators they generate a free subgroup of rank 4. The term ‘almost every’ is made precise by using definitions borrowed from the theory of entire functions regarding densities of subsets.
The authors go on to compute the real cohomology of \(\mathcal J(\mathbb{Z})\) with uniformly constant support and show that this is naturally isomorphic to the cohomology of the nilpotent part of the Witt algebra. In order to compute the cohomology the authors study the coset space \(\mathcal J(\mathbb{R})/\mathcal J(\mathbb{Z})\), and prove various topological and geometric results about this space.

MSC:

20E18 Limits, profinite groups
58H10 Cohomology of classifying spaces for pseudogroup structures (Spencer, Gelfand-Fuks, etc.)
13F25 Formal power series rings
20E07 Subgroup theorems; subgroup growth
20F05 Generators, relations, and presentations of groups

Citations:

Zbl 0977.20020
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