"Geometry and Analysis on Groups" Research Seminar
On the isomorphism problem for (hyperbolic) one-relator groups.
Alan Logan (Heriot-Watt)
In the 1960s Magnus conjectured that two one-relator groups \(\langle x; R\rangle\) and \(\langle y; S\rangle\) are isomorphic if and only if they "obviously" are. Counter-examples were found to this conjecture in the 1970s. Subsequent progress on the isomorphism problem for these groups has considered special cases and no "unified" approach has been suggested. In contrast, Gromov hyperbolic groups have been shown to have decidable isomorphism problem via a unified approach, based on JSJ-decompositions of groups.
This talk is based on joint work with Giles Gardam and Dawid Kielak. I will unpack the word "obviously" in the previous paragraph, and I will explain some known counter-examples to the conjecture. We shall then explore the isomorphism problem for hyperbolic one-relator groups, focusing on the following two questions:
(1) Does the Magnus conjecture hold for hyperbolic one-relator groups?
(2) What does the JSJ-decomposition of a hyperbolic one-relator group look like?