Abstracts for MPI-Oberseminar

An unprovable theorem is a theorem about basic mathematical objects that can only be proved using more than the usual axioms for mathematics (ZFC = Zermelo Frankel set theory with the Axiom of Choice) - and that has been proved using standard extensions of ZFC generally adopted in the mathematical logic community. The highlight of the talk is the presentation of a new unprovable theorem concerning the structure of kernels in discrete and finite directed graphs. We first review some previous examples of unprovable theorems, as time permits.

The free Burnside group of exponent n, B(r,n), is the quotient of the free group of rank r by the subgroup generated by all n-th powers. This group was introduced in 1902 by W. Burnside who asked wether it has to be finite or not. Since the work of P.S. Novikov and S.I. Adian in the late sixties, it is knows that for exponent n large enough the answer is no. In this talk, we are interested in the outer automorphisms of B(r,n). Using a geometrical formulation of the small cancellation theory, we explore the following questions : Which elements of Out(B(r,n)) have infinite order?

For groups with finitely many generators and relators, one can always obtain a minimal group presentation by deleting redundant relators. I will show that such a process does not apply to infinitely presented groups by exhibiting a finitely generated group with no minimal group presentations. The counterexample I will present is moreover 4-soluble and enjoys very special features within the space of marked groups. I will then investigate where soluble groups are located in the space of marked groups, depending on whether they are finitely presented or not.