On the codimension growth of 𝐺-graded algebras

Abstract

Let W W be an associative PI-affine algebra over a field F F of characteristic zero. Suppose W W is G G -graded where G G is a finite group. Let exp ⁡ ( W ) \exp (W) and exp ⁡ ( W e ) \exp (W_{e}) denote the codimension growth of W W and of the identity component W e W_{e} , respectively. We prove exp ⁡ ( W ) ≤ | G | 2 exp ⁡ ( W e ) . \exp (W)\leq |G|^2 \exp (W_{e}). This inequality had been conjectured by Bahturin and Zaicev.