Product separability in central extensions
Lawk Mineh
TL;DR
The paper investigates how separability properties behave under central extensions of hyperbolic groups, focusing on product separability and related double coset and nilpotent-product stability. It shows that a central extension $1 o Z o G o Q o 1$ with $Q$ a subgroup separable locally quasiconvex hyperbolic group is product separable when $G$ is subgroup separable, and proves that central extensions of double coset separable groups by finitely generated $Z$ are double coset separable iff subgroup separable, with direct products by finitely generated nilpotent groups preserving double coset separability. The work introduces bottlenecked product representatives and establishes a main theorem: if $G$ is a central extension of a product separable group $Q$ by a finitely generated $Z$ and $Q$ has bottlenecked product representatives over finitely generated subgroups, then $G$ is product separable. These results yield corollaries for hyperbolic limit groups and circle bundles, highlighting a strong link between negative curvature and separability properties and broadening the class of known product separable groups.
Abstract
We show that a central extension of locally quasiconvex subgroup separable hyperbolic group is product separable, so long as it is subgroup separable. We also establish that a central extension of a double coset separable group by a finitely generated group is double coset separable if and only if it is subgroup separable, and that double coset separability is stable under taking direct products with finitely generated nilpotent groups.