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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.

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    $\begingroup$ Not every finitely generated group of subexponential growth isresidually finite (Erschler). The answer to the first question is not known. $\endgroup$ – Mark Sapir Feb 18 '13 at 17:15
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    $\begingroup$ @Stefan: you could maybe edit the question accordingly. On the other hand, it remains an open question whether there exists a nontrivial f.g. group of subexponential growth with no nontrivial finite quotient. (The derived subgroups of full topological groups of infinite minimal subshifts, which are infinite, f.g., amenable and simple, have free subsemigroups by Matui-2011 so have an exponential growth). $\endgroup$ – YCor Feb 18 '13 at 20:08
  • $\begingroup$ @Ives: I have edited the question, providing the reference to Erschler's result. $\endgroup$ – Stefan Kohl Feb 18 '13 at 20:49
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    $\begingroup$ Btw here's part of the argument for Erschler's result: take for granted that there exists a f.g. group $G$ with a central subgroup $Z$ which is an infinite-dimensional vector space over the field on $p$ elements, such that $G/Z$ has subexponential growth (actually in the example, $p=2$ and $G$ is the 1st Grigorchuk group). If $Z'$ is a hyperplane of $Z$ s.t. $G/Z'$ is residually finite, then there exists a normal subgroup of finite index $N$ such that $Z'=N\cap Z$. There are countably many such $N$ and hence such $Z'$. On the other hand, there are uncountably many $Z'$, so one $G/Z'$ isn't RF. $\endgroup$ – YCor Feb 18 '13 at 21:05
  • $\begingroup$ Some evidence for: arxiv.org/abs/1105.0719 $\endgroup$ – Ian Agol Feb 19 '13 at 2:36
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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, arXiv, 2016.

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