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Summary: THE TRUE PROSOLUBLE COMPLETION OF A GROUP
EXAMPLES AND OPEN PROBLEMS
Goulnara Arzhantseva, Pierre de la Harpe, and Delaram Kahrobaei
August, 2006
Abstract. The true prosoluble completion PS() of a group is the inverse limit of the
projective system of soluble quotients of . Our purpose is to describe examples and to
point out some natural open problems. We discuss a question of Grothendieck for pro nite
completions and its analogue for true prosoluble and true pronilpotent completions.
1. Introduction.
2. Completion with respect to a directed set of normal subgroups.
3. Universal property.
4. Examples of directed sets of normal subgroups.
5. True prosoluble completions.
6. Examples.
7. On the true prosoluble and the true pronilpotent analogues of Grothendieck's problem.
Goulnara Arzhantseva and Pierre de la Harpe, Section de Math ematiques, Universit e
de Gen eve, C.P. 64, CH-1211 Gen eve 4, Suisse.
E-mail: Goulnara.Arjantseva@math.unige.ch, Pierre.delaHarpe@math.unige.ch
Delaram Kahrobaei, Mathematical Institute (Room 315), University of St Andrews,
North Haugh, St Andrews, Fife, KY16 9SS, Scotland.
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