Spectral theory on a free group and algebraic curves.

*(English)*Zbl 0583.60068This is an important paper with beautiful results but unfortunately it is difficult to read (especially § 1) - there are many unprecise formulations and a lot of misprints. The author considers random walks on a finitely generated free group defined by a probability p of finite support. The first main result states that (under some mild extra conditions) the associated Green functions are algebraic.

If the probability p is concentrated on the generators of the free group the author obtains more detailed results on the structure of the Green functions. Finally if p is also symmetric precise information is given about the spectrum and the eigenfunction expansion of the associated operator (to p) on \(l^ 2.\)

Very similar results where obtained independently by T. Steger in his thesis (St. Louis, Missouri, 1985) using different methods.

If the probability p is concentrated on the generators of the free group the author obtains more detailed results on the structure of the Green functions. Finally if p is also symmetric precise information is given about the spectrum and the eigenfunction expansion of the associated operator (to p) on \(l^ 2.\)

Very similar results where obtained independently by T. Steger in his thesis (St. Louis, Missouri, 1985) using different methods.

Reviewer: P.Gerl

##### MSC:

60G50 | Sums of independent random variables; random walks |