On Shalev's conjecture for type 𝐴�𝑛� and 2𝐴�𝑛�

Abstract

In the paper, we consider images of finite simple projective special linear and unitary groups under power words. In particular, we show that, if G≃PSLεn(q), then, for every power word of type xM, there exist constants c and N such that |ω(G)|>cln(n)|G|n whenever |G|>N.