2020/05/01 by Simon André, Jonathan Fruchter
Computer Science · Mathematics · #Abelian group #Advanced Operator Algebra Research #Combinatorics #Computer science #Conjecture #Corollary #Economics #Extension (predicate logic) #Finite Group Theory Research #Free product #Geometric and Algebraic Topology #Geometry #Group (periodic table) #Hyperbolic function #Hyperbolic group #Hyperbolic manifold #Mathematical analysis #Mathematics #Order (exchange) #Physics #Product (mathematics) #Pure mathematics #Relatively hyperbolic group #math.GR #math.LO #semigroups and automata theory
paper · pdf · doi:10.1112/jlms.12526
57 pages, no figures; added references, revised argument in subsection 3.4, results unchanged
openalex publication_date 2020/05/01 · arxiv created 2020/05/23 · openalex created_date 2022/02/24 · arxiv updated 2022/03/09 · openalex updated_date 2026/08/05
We generalise Merzlyakov's theorem about the first-order theory of non-abelian free groups to all acylindrically hyperbolic groups. As a corollary, we deduce that if G is an acylindrically hyperbolic group and E(G) denotes the unique maximal finite normal subgroup of G, then G and the HNN extension G∗E(G), which is simply the free product G∗ℤ when E(G) is trivial, have the same ∀∃-theory. As a consequence, we prove the following conjecture, formulated by Casals-Ruiz, Garreta and de la Nuez González: acylindrically hyperbolic groups have trivial positive theory. In particular, one recovers a result proved by Bestvina, Bromberg and Fujiwara, stating that, with only the obvious exceptions, verbal subgroups of acylindrically hyperbolic groups have infinite width.