GammaSqueeze
Using this deep observation of Andre Weil, SL solved a question of Gilbert Baumslag that had been open for over 30 years as a small part of his PhD thesis
The Question was raised in page 121 of the book:
The History of Combinatorial Group Theory: A Case Study in the History of Ideas, by Chandler and Magus.
The moral of the story is that if a family of groups is very much like free group, as in the case of Parafree groups, then to discriminate members of such a family of groups you might try investigating topological properties of their representation varieties over an algebraic group where free groups embed, even if you don't know if the corresponding representation varieties are non-singular as is the case for the representation variety of a fg free group in SL(2,C). Using Andre Weil's deep observation we can just count points over a finite field of P elements, for a suitably chosen prime integer P, after reducing mod P the defining polynomials of the corresponding representation varieties of the two groups we are attempting to discriminate. This was a fruitful approach and quite amenable to computations using packages such a Singular, Gap, or the Computational Algebra System Magma... Etc.
SL gives a talk here on the representation varieties of Parafree Groups : youtu.be/rLL9IKoh0ms
Some basic facts on Parafree groups here:
Using this deep observation of Andre Weil, SL solved a question of Gilbert Baumslag that had been open for over 30 years as a small part of his PhD thesis
The Question was raised in page 121 of the book:
The History of Combinatorial Group Theory: A Case Study in the History of Ideas, by Chandler and Magus.
The moral of the story is that if a family of groups is very much like free group, as in the case of Parafree groups, then to discriminate members of such a family of groups you might try investigating topological properties of their representation varieties over an algebraic group where free groups embed, even if you don't know if the corresponding representation varieties are non-singular as is the case for the representation variety of a fg free group in SL(2,C). Using Andre Weil's deep observation we can just count points over a finite field of P elements, for a suitably chosen prime integer P, after reducing mod P the defining polynomials of the corresponding representation varieties of the two groups we are attempting to discriminate. This was a fruitful approach and quite amenable to computations using packages such a Singular, Gap, or the Computational Algebra System Magma... Etc.
SL gives a talk here on the representation varieties of Parafree Groups : youtu.be/rLL9IKoh0ms
Some basic facts on Parafree groups here: