Giovedì 4 settembre 2025 dalle ore 10:00, nel quadro del ciclo di seminari Al@Bicocca, Jim Belk (University of Glasgow) e Collin Bleak (University of St. Andrews) terranno i seguenti interventi
Relatore: Jim Belk
Titolo: Boone-Higman embeddings of Aut(F_n) and mapping class groups of punctured surfaces
Abstract: The Boone-Higman conjecture asserts that every finitely presented
group with solvable word problem embeds into a finitely presented simple group.
Such embeddings are now known for many classes of finitely presented groups,
including arithmetic groups, right-angled Artin groups, Coxeter groups, hyper-
bolic groups, self-similar groups, Baumslag-Solitar groups, and free-by-cyclic
groups. This talk will survey these results and then discuss some recent work with
Francesco Fournier-Facio, James Hyde, and Matt Zaremsky that embeds each
Aut(Fn) into a finitely presented simple group. This also yields Boone-Higman
embeddings for braid groups and many of their generalizations, including map-
ping class groups of punctured surfaces and several families of Artin groups.
Relatore: Collin Bleak
Titolo: Strong Generation in Simple Vigorous Group
The simple vigorous groups are a broad class of groups of homeomor-
phisms of Cantor space that includes Thompson’s group V , its various generalisa-
tions and many others such as Nekrashevych’s groups of dynamical origin. Bleak,
Elliott and Hyde (2024) proved that every finitely generated simple vigorous group
is 2-generated, and, in this talk, we give several strong generation results for this
class of simple groups. For example, we prove that if G is a finitely generated
simple vigorous group, then G is generated by three involutions, that G is gener-
ated by an element of order m and an element of order n for any choice of m ≥ 2
and n ≥ 3, that G has a minimal generating set of size k for all k ≥ 2, that every
non-trivial element of G is contained in a generating pair and that the direct power
Gn is 2-generated for all n. These results are analogous to well-known results
for finite simple groups, but of course the proofs in this context are quite different.
One consequence of our results is that Thompson’s group V is (2, 3)-generated,
which answers a question of Sapir (2017). Another consequence is that every
finitely generated group quasi-isometrically embeds in a (2, 3)-generated simple
group, generalising theorems of Hall (1974) and Bridson (1998). Joint with Casey
Donoven, Scott Harper, and James Hyde.
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