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.
