Malnormality is undecidable in hyperbolic groups (Q5951497)

A subgroup \(H\) of a group \(G\) is malnormal if for every \(g\in G\setminus H\) it is the case that \(g^{-1}Hg\cap H=1\). The authors answer a question of Myasnikov by proving among other things that there exists a torsion-free hyperbolic group \(G\) for which there is no algorithm to decide which finitely generated subgroups are malnormal. The group \(G\) can be arranged to be the fundamental group of a compact 2-dimensional complex that is negatively curved (in the sense of Alexandrov). The result is obtained using a construction due to \textit{E. Rips} [Bull. Lond. Math. Soc. 14, 45-47 (1982; Zbl 0481.20020)].