Just-infinite branch pro-p groups introduced earlier by the author are shown to be rigid on trees obtained by deleting levels, forcing every branch action on the p-adic tree to be zero-dimensional and disproving Boston's conjecture.
Arboreal representations of linear groups
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Ab\'ert and Vir\'ag conjectured in 2005 that any embedding of a linear group over a pro-$p$ domain into the group of $p$-adic automorphisms $W_p$ should be zero-dimensional. We prove their conjecture in greater generality, namely for embeddings of linear groups over any integral domain into the automorphism group of a bounded rooted tree.
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Weakly branch actions: first-order theory, rigidity and Boston's conjecture
Just-infinite branch pro-p groups introduced earlier by the author are shown to be rigid on trees obtained by deleting levels, forcing every branch action on the p-adic tree to be zero-dimensional and disproving Boston's conjecture.