Abstract
Suppose is a simple and simply connected algebraic group over an algebraic number field and is a finite set of valuations of containing all Archimedean valuations. This paper is a study of the connections between abstract properties of the -arithmetic subgroup and the congruence property, i.e. the finiteness of the corresponding congruence kernel . In particular, it is shown that if the profinite completion satisfies condition (PG), (i.e., for any integer > 0 and any prime there exist and such that for all ), then is finite. Examples are given demonstrating the possibility of effectively verifying (PG)' .
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