arXiv subjects

Benjamin Bachner

Publications and source records attributed to Benjamin Bachner.

3 recordsLinked to original sources

arXiv2026

Linear sofic representations of amenable algebras

We study the notion of linear sofic approximations for algebras, analogous to the concept of sofic representations for groups. We prove that for a finitely generated amenable $K$-algebra with no zero divisors, all linear sofic representations are conjugate. This provides an algebraic analogue to Elek and Szab\'o's theorem for amenable groups. The proof relies on a "linear monotiling" technique, constructed using a theorem by Bre\v{s}ar, Meshulam and \v{S}emrl on locally linearly dependent operators. Finally, we apply this uniqueness result to the problem of weak stability in the rank metric, showing that the group algebra of an amenable group is weakly stable if and only if the group is residually finite.

math.RA↗

arXiv2025

On $L^1$-approximation of groups

A longstanding open problem in the intersection of group theory and operator algebras is whether all groups are MF, that is, approximated by asymptotic representations with respect to the operator norm. More generally, for $1 \leq p \leq \infty$, it has been asked by Thom in his ICM address whether there exist groups which are not approximated with respect to the Schatten $p$-norm. The cases of $1 < p < \infty$ were addressed in previous works. We settle the case $p=1$, solving a question left open by Lubotzky and Oppenheim.

math.GR↗

arXiv2024

Uniform rank metric stability of Lie algebras and groups

We study uniform stability of discrete groups, Lie groups and Lie algebras in the rank metric, and the connections between uniform stability of these objects. We prove that semisimple Lie algebras are far from being flexibly $\mathbb{C}$-stable, and that semisimple Lie groups and lattices in semisimple Lie groups of higher rank are not strictly $\mathbb{C}$-stable. Furthermore, we prove that free groups are not uniformly flexibly $F$-stable over any field $F$.

math.GR↗