Is it true that every finitely generated infinite simple group has
exponential (word-)growth?

Remark: As Mark Sapir has pointed out, the question whether
every finitely generated group of subexponential growth is even residually finite
has been answered in the negative in

Anna Erschler. Not residually finite groups of intermediate growth,
commensurability and non-geometricity, <em>J. Algebra</em> 272 (2004), no. 1, 154–172,
<http://www.sciencedirect.com/science/article/pii/S0021869303006410>.