We develop a notion of degree for functions between two abelian groups that allows us to generalize the Chevalley Warning Theorems from fields noncommutative rings or prime power order.

We show that for integers k ≥ 2 and n ≥ 3, the diameter of the Cayley graph of SLn(Z/kZ) associated to a standard two-element generating set, is at most a constant times n ln k. This answers a question of A. Lubotzky concerning SLn(Fp) and is unexpected because these Cayley graphs do not form an expander family. Our proof amounts to a quick algorithm for finding short words representing element...

We prove that if $$G(R)=G_\pi (\Phi ,R)$$ $$(E(R)=E_{\pi }(\Phi , R))$$ is an (elementary) Chevalley group of rank $$> 1$$ R a local ring (with $$\frac{1}{2}$$ for the root systems $${{\textbf{A}}}_2, {{\textbf{B}}}_l, {{\textbf{C}}}_l, {{\textbf{F}}}_4, {{\textbf{G}}}_2$$ and with $$\frac{1}{3}$$ $${{\textbf{G}}}_{2})$$ then G(R) (or (E(R)) regularly bi-interpretable R. As consequence this the...

This paper extends the comparison results for cohomological support varieties for Chevalley groups of the form G(Fp) established by J. Carlson, Z. Lin, and D. Nakano to Chevalley groups of the form G(Fpd). Other topics considered are complexity and non-maximal support varieties for modules. Techniques developed by J. Pevtsova and the author, together with the use of Weil restriction functors, a...