Formal solutions and the first-order theory of acylindrically hyperbolic groups
| Autor: | André, Simon, Fruchter, Jonathan |
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| Rok vydání: | 2020 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1112/jlms.12526 |
| Popis: | 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 $G\dot{\ast}_{E(G)}$, which is simply the free product $G\ast\mathbb{Z}$ when $E(G)$ is trivial, have the same $\forall\exists$-theory. As a consequence, we prove the following conjecture, formulated by Casals-Ruiz, Garreta and de la Nuez Gonz\'alez: 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. Comment: 57 pages, no figures; added references, revised argument in subsection 3.4, results unchanged |
| Databáze: | arXiv |
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