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.
On the fixed-point proportion of self-similar groups
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We prove that super strongly fractal groups acting on regular rooted trees have null fixed-point proportion. In particular, we show that the fixed-point proportion of an infinite family of iterated monodromy groups of exceptional complex polynomials have the same property. The proof uses the approach of Rafe Jones in [15] based on martingales and a recent result of the first author on the dynamics of self-similar groups [6].
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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.