Lie Dimension Subrings

We compare, for L a Lie ring over the integers, its lower central series (\gamma_n(L))_{n>0} and its dimension series defined by \delta_n(L):=L\cap \varpi^n(L) in the universal enveloping algebra of L. We show that \gamma_n(L)=\delta_n(L) for all n<4, but give an example showing that they may differ if n=4. We introduce simplicial methods to describe these results, and to serve as a possible tool for further study of the dimension series.Comment: Small typos fixed wrt v

Lie Algebras and the Dimension Problem

The dimension subgroup problem and the Ore conjecture are two group theoretical problems. In this thesis, translations of these problems to Lie algebras are analyzed and partially solved.2021-11-1

Group and Lie algebra filtrations and homotopy groups of spheres

We establish a bridge between homotopy groups of spheres and commutator calculus in groups, and solve in this manner the “dimension problem” by providing a converse to Sjogren’s theorem: every abelian group of bounded exponent can be embedded in the dimension quotient of a group. This is proven by embedding for arbitrary , the torsion of the homotopy group () into a dimension quotient, via a result of Wu. In particular, this invalidates some long-standing results in the literature, as for every prime , there is some -torsion in 2(2) by a result of Serre. We explain in this manner Rips’s famous counterexample to the dimension conjecture in terms of the homotopy group 4(2) = ℤ∕2ℤ. We finally obtain analogous results in the context of Lie rings: for every prime there exists a Lie ring with -torsion in some dimension quotient