** "Geometry and Analysis on Groups" Research Seminar **

Concretely, we will introduce arithmetically-free gradings of (a broad family) of algebras by (arbitrary) groups, and show:

- If the support \(X\) of the grading is arithmetically-free, then \(A\) is nilpotent of \(|X|\)–bounded class.
- If \(X\) is not arithmetically-free, then it supports the grading of a non-nilpotent algebra.
- If \(X\) is arithmetically-free and admits a good-ordering, then a Lie algebra \(L\) supported by \(X\) is nilpotent of class at most \(|X|^{2^{|X|}}\).

The proof for 1. is combinatorial in nature and is based on an existence result by G. Higman in the special case \((G,\cdot) = (\mathbb{Z}_p,+)\). (It can also be stated in terms of walks in Cayley-graphs.) The proof for 3. uses some Lie theory and touches on several problems of Erdös in additive combinatorics. We conclude with some brief remarks about the connection between arithmetically-free gradings, periodic transformations, and the co-class conjectures for \(p\)–groups.