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.
Random subgroups of branch groups
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abstract
We show that independent Haar-random elements in a super strongly fractal branch profinite group generate a free subgroup acting freely on the boundary of the tree. This improves a previous result of Ab\'ert (2005) for weakly branch profinite groups, where independent random elements were shown to generate free subgroups acting only almost freely on the boundary. Our result also generalizes the analogous result of Ab\'ert and Vir\'ag (2005) for iterated wreath products.
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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.