Ultrametric analogues of Ulam stability of groups

We study stability of metric approximations of countable groups with respect to groups endowed with ultrametrics, the main case study being a $p$-adic analogue of Ulam stability, where we take $\mathrm{GL}_n(\mathbb{Z}_p)$ as approximating groups instead of $\mathrm{U}(n)$. For finitely presented groups, the ultrametric nature implies equivalence of the pointwise and uniform stability problems, and the profinite one implies that the corresponding approximation property is equivalent to residual finiteness. Moreover, a group is uniformly stable if and only if its largest residually finite quotient is. We provide several examples of uniformly stable groups, including finite groups, virtually free groups, some groups acting on rooted trees, and certain lamplighter and (Generalized) Baumslag--Solitar groups. We construct a finitely generated group that is not uniformly stable. Finally, we prove and apply a (bounded) cohomological criterion for stability of a finitely presented group.Comment: 86 pages. v2: added Section 8 with non-examples, improved the (bounded) cohomological criterion, developed the quantitative aspects. v3: minor change

Ulam stability of lamplighters and Thompson groups

We show that a large family of groups is uniformly stable relative to unitary groups equipped with submultiplicative norms, such as the operator, Frobenius, and Schatten $p$-norms. These include lamplighters $\Gamma \wr \Lambda$ where $\Lambda$ is infinite and amenable, as well as several groups of dynamical origin such as the classical Thompson groups $F, F', T$ and $V$. We prove this by means of vanishing results in asymptotic cohomology, a theory introduced by the second author, Glebsky, Lubotzky and Monod, which is suitable for studying uniform stability. Along the way, we prove some foundational results in asymptotic cohomology, and use them to prove some hereditary features of Ulam stability. We further discuss metric approximation properties of such groups, taking values in unitary or symmetric groups.Comment: 27 pages. v2: final version, to appear in Mathematische Annale

Finitely generated simple left orderable groups with vanishing second bounded cohomology

We prove that the finitely generated simple left orderable groups constructed by the second author with Hyde have vanishing second bounded cohomology, both with trivial real and trivial integral coefficients. As a consequence, these are the first examples of finitely generated non-indicable left orderable groups with vanishing second bounded cohomology. This answers Question 8 from the 2018 ICM proceedings article of Andr\'es Navas.Comment: 18 pages. v2: added affiliations and fixed typo

Hopfian wreath products and the stable finiteness conjecture

We study the Hopf property for wreath products of finitely generated groups, focusing on the case of an abelian base group. Our main result establishes a strong connection between this problem and Kaplansky's stable finiteness conjecture. Namely, the latter holds true if and only if for every finitely generated abelian group $A$ and every finitely generated Hopfian group $\Gamma$ the wreath product $A \wr \Gamma$ is Hopfian. In fact, we characterize precisely when $A \wr \Gamma$ is Hopfian, in terms of the existence of one-sided units in certain matrix algebras over $\mathbb{F}_p[\Gamma]$, for every prime factor $p$ occurring as the order of some element in $A$. A tool in our arguments is the fact that fields of positive characteristic locally embed into matrix algebras over $\mathbb{F}_p$ thus reducing the stable finiteness conjecture to the case of $\mathbb{F}_p$. A further application of this result shows that the validity of Kaplansky's stable finiteness conjecture is equivalent to a version of Gottschalk's surjunctivity conjecture for additive cellular automata.Comment: 28 pages, comments welcome

Aut-invariant quasimorphisms on groups

For a large class of groups, we exhibit an infinite-dimensional space of homogeneous quasimorphisms that are invariant under the action of the automorphism group. This class includes non-elementary hyperbolic groups, infinitely-ended finitely generated groups, some relatively hyperbolic groups, and a class of graph products of groups that includes all right-angled Artin and Coxeter groups that are not virtually abelian. This was known for $F_2$ by a result of Brandenbursky and Marcinkowski, but is new even for free groups of higher rank, settling a question of Mikl\'os Ab\'ert. The case of graph products of finitely generated abelian groups settles a question of Michal Marcinkowski. As a consequence, we deduce that a variety of Aut-invariant norms on such groups are unbounded.Comment: 17 page

Local Hilbert--Schmidt stability

We introduce a notion of local Hilbert--Schmidt stability, motivated by the recent definition by Bradford of local permutation stability, and give examples of (non-residually finite) groups that are locally Hilbert--Schmidt stable but not Hilbert--Schmidt stable. For amenable groups, we provide a criterion for local Hilbert--Schmidt stability in terms of group characters, by analogy with the character criterion of Hadwin and Shulman for Hilbert--Schmidt stable amenable groups. Furthermore, we study the (very) flexible analogues of local Hilbert--Schmidt stability, and we prove several results analogous to the classical setting. Finally, we prove that infinite sofic, respectively hyperlinear, property (T) groups are never locally permutation stable, respectively locally Hilbert--Schmidt stable. This strengthens the result of Becker and Lubotzky for classical stability, and answers a question of Lubotzky.Comment: 29 pages, comments welcom

Median quasimorphisms on CAT(0) cube complexes and their cup products

Cup products provide a natural approach to access higher bounded cohomology groups. We extend vanishing results on cup products of Brooks quasimorphisms of free groups to cup products of median quasimorphisms, i.e., Brooks-type quasimorphisms of group actions on CAT(0) cube complexes. In particular, we obtain such vanishing results for groups acting on trees and for right-angled Artin groups. Moreover, we outline potential applications of vanishing results for cup products in bounded cohomology.Comment: 36 pages, 12 figures; v2: corrected mistake in the first version related to infinite staircases. v3: final version, to appear in Geometriae Dedicat