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, *J. Algebra* 272 (2004), no. 1, 154–172,
<http://www.sciencedirect.com/science/article/pii/S0021869303006410>.



Is it true that every f.g. infinite simple group has exponential growth?
No, there exists a finitely generated infinite simple group of intermediate 
growth. This has meanwhile been found out by Volodymyr Nekrashevych, cf.
[*Palindromic subshifts and simple periodic groups of intermediate growth*](http://arxiv.org/pdf/1601.01033.pdf),
arXiv, 2016.


