Event № 237
I will present an elementary proof of the following theorem of Alexander Olshanskii:
Let F be a free group and let A,B be finitely generated subgroups of infinite index in F. Then there exists an infinite index subgroup C of F which contains both A and a finite index subgroup of B.
The proof is carried out by introducing a 'profinite' measure on the discrete group F, and is valid also for some groups which are not free.Some applications of this result will be discussed:
1. Group Theory - Construction of locally finite faithful actions of countable groups.
2. Number Theory - Discontinuity of intersections for large algebraic extensions of local fields.
3. Ergodic Theory - Establishing cost 1 for groups boundedly generated by subgroups of infinite index and finite cost.