POLYNOMIAL INDEX GROWTH GROUPS

Abstract

We show that the order of a finite simple of Lie type is bounded by a small constant power of its exponent. This confirms, in a strengthened form, a conjecture of Vaughan-Lee and Zel'manov on the order and exponent of almost simple groups. We also obtain various structural restrictions on groups of polynomial index growth. Combining the above results we construct finitely generated residually finite groups of polynomial index growth which are neither linear nor boundedly generated. This answers questions of Segal and Platonov–Rapinchuk respectively. A further question of Platonov–Rapinchuk concerning a weakened polynomial index growth assumption is also answered.