Remarks on profinite groups having few open subgroups

Examples are given of profinite groups that are not strongly complete, and have other `bad' properties, yet have only finitely many open subgroups of each finite index. It is shown that a profinite group with the latter property must be finite if it has finite exponent. The problem of characterizing strongly complete groups in terms of their power subgroups is discussed.Comment: The paper has been rewritten and expanded, with some new material. Similar examples appear in N. Nikolov: Algebraic properties of profinite groups, arXiv:1108.5130 and in the earlier versions of this not

On finitely generated profinite groups I: strong completeness and uniform bounds

We prove that in every finitely generated profinite group, every subgroup of finite index is open; this implies that the topology on such groups is determined by the algebraic structure. This is deduced from the main result about finite groups: let $w$ be a `locally finite' group word and $d\in\mathbb{N}$. Then there exists $f=f(w,d)$ such that in every $d$-generator finite group $G$, every element of the verbal subgroup $w(G)$ is equal to a product of $f$ $w$-values. An analogous theorem is proved for commutators; this implies that in every finitely generated profinite group, each term of the lower central series is closed. The proofs rely on some properties of the finite simple groups, to be established in Part II.Comment: 66 page

On finitely generated profinite groups II, products in quasisimple groups

We prove two results. (1) There is an absolute constant $D$ such that for any finite quasisimple group $S$, given 2D arbitrary automorphisms of $S$, every element of $S$ is equal to a product of $D$ `twisted commutators' defined by the given automorphisms. (2) Given a natural number $q$, there exist $C=C(q)$ and $M=M(q)$ such that: if $S$ is a finite quasisimple group with $| S/\mathrm{Z}(S)| >C$, $\beta_{j}$ $(j=1,...,M)$ are any automorphisms of $S$, and $q_{j}$ $(j=1,...,M)$ are any divisors of $q$, then there exist inner automorphisms $\alpha_{j}$ of $S$ such that $S=\prod_{1}^{M}[S,(\alpha_{j}\beta_{j})^{q_{j}}]$. These results, which rely on the Classification of finite simple groups, are needed to complete the proofs of the main theorems of Part I.Comment: 34 page

Defining R and G(R)

We show that for Chevalley groups G(R) of rank at least 2 over a ring R the root subgroups are essentially (nearly always) the double centralizers of corresponding root elements. In very many cases this implies that R and G(R) are bi-interpretable, yielding a new approach to bi-interpretability for algebraic groups over a wide range of rings and fields. For such groups it then follows that the group G(R) is finitely axiomatizable in the appropriate class of groups provided R is finitely axiomatizable in the corresponding class of rings.Comment: (1) New Theorem 1.1 generalizes earlier main theorems.(2) New version incorporates content of arXiv:2007.11440 (3) Latest version has small corrections. To appear in J. Eur. Math. So

Finitely generated groups with polynomial index growth

We prove that a finitely generated soluble residually finite group has polynomial index growth if and only if it is a minimax group. We also show that if a finitely generated group with PIG is residually finite-soluble then it is a linear group. These results apply in particular to boundedly generated groups; they imply that every infinite BG residually finite group has an infinite linear quotient.Comment: To appear in Crelle's Journa

The Long-Term Effects of Cross-Listing Investor Recognition, and Ownership Structure on Valuation

The authors show that the widening of a foreign firm's U.S. investor base and the improved information environment associated with cross-listing on a U.S. exchange each have a separately identifiable effect on a firm's valuation. The increase in valuation associated with cross-listing is transitory, not permanent. Valuations of Canadian firms peak in the year of cross-listing and fall monotonically thereafter, regardless of the level of U.S. investor holdings or the ownership structure of the firm. Cross-listed firms with a 20 per cent or more blockholder attract a similar number of U.S. institutional investors as widely held firms, on average, but experience a lower increase in valuation at high levels of investor recognition. While U.S. investors are less willing to invest in firms with dual-class shares, these firms benefit more from cross-listing even when they fail to widen their U.S. investor base, suggesting that the reduction in information asymmetry between controlling and minority investors has a separate impact on valuation for firms where agency problems are greatest.Financial markets; International topics

On the finite axiomatizability of some metabelian profinite groups

Finitely generated (non-abelian) free metabelian pro-p groups, and wreath products of f.g. free abelian pro-p groups, are all finitely axiomatizable in the class of all profinite groups

A profinite analogue of Lasserre's theorem

A soluble pro-p group of finite rank is finitely axiomatizable in the class of all profinite groups if and only if for each open subgroup H, the image of Z(H) in the abelianization of H is finite, subject to some suitable hypothesis of finite presentability.Comment: Solves Problem 1 in `Finite axiomatizability for profinite groups', Proc. London Math. Soc.(3) 123, (2021