Embedding free amalgams of discrete groups in non-discrete topological groups

The main result is the theorem: Assume that the set of nontrivial groups {G(mu)}(mu is an element of I) contains either three groups or two groups of which one has order at least 3. Then the free amalgam Omega(1) of the groups G(mu), mu is an element of I, can be embedded in a group G = gp{Omega(1)} in such a way that for each fixed cardinal beta < \G\, G admits a non-discrete Hausdorff topology such that 1) G is a 0-dimensional topological group; 2) every neighbourhood is of cardinality \G\; 3) if a subgroup M of G is conjugate to a subgroup of G(mu) for some mu is an element of I or M is a finite extension of a cyclic group, then M is discrete; 4) every subgroup of G of cardinality gamma less than or equal to beta is discrete; and 5) if beta is finite, then G may be chosen to be metrizable. This result depends on the method of A. Yu. Ol'shanskii. A special case of the theorem, together with applications, is given in [4].