Conjugacy separability of some one-relator groups

We extend Dye's reconstruction theorem, which classifies isomorphisms between full groups, to a classification of homomorphisms between full groups. For full groups of ergodic p.m.p. equivalence relations, our result roughly says that such homomorphisms come only from actions of the equivalence relation, or of one of its symmetric powers. This has several rigidity consequences for homomorphisms between full groups. Our main application is a characterization of property (T) for ergodic p.m.p. equivalence relations purely in full group terms, without using their topology: an ergodic p.m.p. equivalence relation has (T) iff all non-free ergodic boolean actions of its full group are strongly ergodic.

math.GR↗

arXiv2026

The Burness-Giudici Conjecture on Primitive Groups with Socle $Ree(q)$ and $Sz(q)$

Let $G$ be a transitive permutation group on $Ω$ containing two points $α, β$ such that $G_α\cap G_β=1$. The Saxl graph $Σ(G)$ of $(G, Ω)$ is defined as the graph with vertex set $Ω$, where two vertices $α', β'$ are adjacent if and only if $G_{α'}\cap G_{β'}=1$. Burness and Giudici conjectured that for any primitive permutation group $G$, its Saxl graph $Σ(G)$ satisfies the property that any two vertices share a common neighbor. We study this common-neighbor property for primitive groups whose socle is a simple group of Lie type of rank one, namely $PSL(2,q)$, $PSU(3,q)$, $Ree(q)$ or $Sz(q)$. In this paper, we establish the property for groups with socle $Ree(q)$ or $Sz(q)$, except possibly when $soc(G)=Ree(q)\lneq G $ and a point stabilizer $M$ satisfies $M\cap soc(G)\cong Z_2\times PSL(2,q)$.

math.GR↗

arXiv2026

Rings of invariants of three-dimensional upper-triangular representations of finite groups

The work proves that, for finite three-dimensional upper triangular groups over a field of odd characteristic generated by pseudoreflections, maximal unipotent subgroup of which is abelian, the ring of invariants is polynomial if and only if the maximal unipotent subgroup is generated by pseudoreflections or does not contain transvections.

math.GR↗