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.
Bartholdi, Branch rings, thinned rings, tree enveloping rings—Erratum, Israel Journal of Mathematics, 193 (1), 507–508, 2013
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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.