On isomorphisms to a free group and beyond (Q6601472)

The isomorphism problem has been solved for a number of classes of groups, among them finitely generated nilpotent groups (see [\textit{I. Bumagin} et al., J. Pure Appl. Algebra 208, No. 3, 961--977 (2007; Zbl 1121.20026)]) and hyperbolic groups (see [\textit{Z. Sela}, Ann. Math. (2) 141, No. 2, 217--283 (1995; Zbl 0868.57005); \textit{F. Dahmani} and \textit{V. Guirardel}, Geom. Funct. Anal. 21, No. 2, 223--300 (2011; Zbl 1258.20034)] for example). \N\NThe author proves the following two theorems: \N\NTheorem 1: Let \(G\) be a finitely presented group with a given algorithm for solving the word problem in \(G\). Then, for any given \(n\geq 1\) it is algorithmically possible to find out whether or not \(G\) is isomorphic to a free group of rank \(n\).\N\NTheorem 2: Let \(G\) be a finitely presented group with a given algorithm for solving the word problem in \(G\) and let \(H\) be a limit group. Then it is algorithmically possible to find out whether or not \(G\) can be embedded in \(H\). \N\NThe results themselves require the results from [\textit{A. A. Razborov}, Math. USSR, Izv. 25, 115--162 (1985; Zbl 0579.20019); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 48, No. 4, 779--832 (1984)].