Growth of finitely presented groups

 Importance: High ✭✭✭
 Subject: Group Theory
 Keywords: finitely presented growth
 Posted by: mdevos on: November 12th, 2007

\begin{problem} Does there exist a finitely presented group of intermediate growth? \end{problem}

See \Def[Wikipedia's growth of groups]{growth rate (group theory)} for definitions of the basics reguarding growth rate in groups (in particular polynomial and exponential growth rates). A finitely generated group has \emph{intermediate growth} if its growth rate (for every finite generating set) is subexponential but superpolynomial.

Most naturally occuring groups have either polynomial growth (such as ${\mathbb Z}^n$) or exponential growth (such as a free group with rank $n > 1$). Milnor [M] famously asked if there exists a finitely generated group with intermediate growth, and this problem was resolved in the affirmative by Grigorchuk [G]. The groups constructed by Grigorchuk are not finitely presented, thus leaving the above problem.

Expert opinion seems to be that there are no finitely presented groups of intermediate growth.

Bibliography

[G] R. I. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means., Izv. Akad. Nauk SSSR Ser. Mat. 48:5 (1984), 939-985.

[M] J. Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968), 1-7.

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