Limit laws for wide varieties of topological groups
by
Dmitri Shakhmatov
Ehime University, Matsuyama, Japan

A partially ordered set is a pair (D, <= ) consisting of a set D together with a relation <= which is:

(i) reflexive , i.e. d <= d for each d in D, and

(ii) transitive , i.e. d₀ <= d₁ and d₁ <= d₂ implies d₀ <= d₂.

A partially ordered set (D, <= ) is directed provided that for every pair d₁, d₂ in D of elements of D there exists d in D such that d₁ <= d and d₂ <= d. A subset C of a directed set (D, <= ) is caled cofinal in (D, <= ) if for every d in D there exists c in C with d <= c. If (D, <= ) is a directed set, then for every d in D the set D _{>= d}={c in D: d <= c} is cofinal in (D, <= ).

A limit law is a map f:(D, <= ) --> F(X) from a directed set (D, <= ) to a free group F(X) over some set X, and an Abelian limit law is a map f:(D, <= ) --> A(X) from a directed set (D, <= ) to a free Abelian group A(X). If f:(D, <= ) --> F(X) is a limit law, and E is a directed subset of D, then the restriction f|_E:(E, <= |_E) --> F(X) of f to E is again a limit law.

We say that a limit law f:(D, <= ) --> F(X) holds in a topological group G, or that G satisfies law f , if for every homomorphism \pi:F(X) --> G from F(X) to G the directed set {\pi(f(d)):d in D} converges to the identity element e_G of G (which means that for every open set U which contains the identity element e_G there exists some d in D such that \pi(f(c)) in U for all c >= d).

A limit law f:(D, <= ) --> F(X) is called trivial if there exists some d in D such that f(c)=e for all c >= d. Clearly, trivial limit laws hold in every topological group, and so they are of no particular interest.

Given a set L of limit laws, the class V(L) of topological groups in which every law from L holds is closed under opeartions of taking Cartesian products, subgroups and continuous homomorphic images (i.e. V(L) forms what is called a wide variety of topological groups).

We give a survey of recent advances in the theory of limit laws and wide varieties of topological groups they generate, as well as present some new results in this area.

Date received: July 25, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagx-55.