For many groups of finite type, topological finite generation, just-infiniteness and strong completeness are equivalent, and new algorithms classify the groups up to isomorphism on binary and ternary trees.
Ab´ ert, Group laws and free subgroups in topological groups, The Bulletin of the London Mathematical Society, 37 (4), 525–534, 2005
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Groups of finite type: classification and structural properties
For many groups of finite type, topological finite generation, just-infiniteness and strong completeness are equivalent, and new algorithms classify the groups up to isomorphism on binary and ternary trees.