On Bartholdi's Conjecture on the Closures of the Aleshin and Bellaterra Groups

In an Oberwolfach report of 2006, L. Bartholdi conjectured that the closures of the Aleshin automaton group \(G(\mathcal{A})\) and of the Bellaterra automaton group \(G(\mathcal{B})\) in the automorphism group of the binary rooted tree are \(\mathbb{Z}_2\)-analytic, and that the closure of \(G(\mathcal{B})\) is a pro-\(2\) Sylow subgroup of \(\mathrm{SL}_2(\mathbb{Z}_2)\). We show that the conjecture is false: the closure of the Bellaterra group is 2-adic analytic but is not isomorphic to a pro-2 Sylow subgroup of \(\mathrm{SL}_2(\mathbb{Z}_2)\) (nor of \(\mathrm{PSL}_2(\mathbb{Z}_2)\) or \(\mathrm{PGL}_2(\mathbb{Z}_2)\)), and the closure of the Aleshin group is not 2-adic analytic. The literal Sylow statement fails for an elementary reason: the closure of \(G(\mathcal{B})\) contains three distinct involutions, \(\mathrm{SL}_2(\mathbb{Z}_2)\) only one. The substance is in two theorems. The closure of \(G(\mathcal{B})\) is isomorphic to \(\{g\)\({}\in\mathrm{GL}_2(\mathbb{Z}_2)\)\({}:g\)\({}\equiv1\bmod2,\ \det g\)\({}=\pm1\}\)\(/\{\pm1\}\), an open subgroup of index \(12\) of \(\mathrm{PGL}_2(\mathbb{Z}_2)\); the proof uses a Möbius model of the Bellaterra automaton due to Kharlampovich and Vdovina (an unrefereed 2026 preprint whose acknowledgement declares AI assistance; all identities taken from it are verified again here), a lifting lemma for congruence subgroups, and a computation in groups of orders \(32\) and \(256\). The closure of \(G(\mathcal{A})\) has index \(2\) in the closure of the group generated by \(G(\mathcal{B})\) and the complement map, which in the Möbius coordinate is not differentiable at the fixed point of a prefix contraction; \(p\)-adic Lie theory then excludes analyticity. No printed answer to the question was found in the sources searched; two works citing the report could not be read. This is an unrefereed note.