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I came across this rather week small cancellation condition $C'\left(\frac{5}{11}\right)$ of a group $G$. It has been proved that $C'\left(\frac16\right)$ is enough for $G$ to contain free subgroups. I was therefore wondering if $\frac{5}{11}$ is maybe enough to still have exponential growth.

Does anyone know of any related papers or results?

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    $\begingroup$ Doesn't $\mathbb{Z}^2$ satisfy $C'(5/11)$? Or have I misremembered the definition? (You should probably give the definition.) $\endgroup$
    – HJRW
    Oct 10, 2012 at 11:05
  • $\begingroup$ @HW: Yes, of course: $5/11 > 1/4$. $\endgroup$
    – user6976
    Oct 10, 2012 at 12:39
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    $\begingroup$ Mark - I agree with your inequality! Hence my wanting to check I'd remembered the definition correctly. $\endgroup$
    – HJRW
    Oct 10, 2012 at 12:51
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    $\begingroup$ The question should probably be about 2/11 which is between 1/5 and 1/6. $\endgroup$
    – Denis Osin
    Oct 10, 2012 at 15:24
  • $\begingroup$ Many thanks for your helpful comments! Denis, would it have exponential growth if it was 2/11? $\endgroup$
    – Elisabeth Fink
    Oct 11, 2012 at 10:29

1 Answer 1

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Every finitely presented group has presentation satisfying $C'(1/5)$. Note that $1/5 < 5/11$. See the book by Olshanskii's book "Geometry of defining relations of groups".

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    $\begingroup$ It is A.I.Golberg's theorem (1978). The proof can be found in Olshanskii's book, but not in the book of Lyndon and Schupp (1977), I think. $\endgroup$
    – Anton Klyachko
    Oct 10, 2012 at 14:27
  • $\begingroup$ @Anton: You are right! $\endgroup$
    – user6976
    Oct 10, 2012 at 15:02
  • $\begingroup$ Many thanks for your answer. The groups I am looking at are infinitely presented. Are there any partial results known for such cases? $\endgroup$
    – Elisabeth Fink
    Oct 11, 2012 at 10:51
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    $\begingroup$ If the group is not finitely presented, the situation is different. At least you know, by Gromov's theorem, that the group cannot have polynomial growth. I do not know if Grigorchuk group or other groups of intermediate growth can have $C'(5/11)$-presentation. You can check the Lysenok presentation of Grigorchuk group. $\endgroup$
    – user6976
    Oct 11, 2012 at 11:48
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    $\begingroup$ It seems that Golberg's theorem remains valid for non-finitely presented groups. Just check the proof in Olshanskii's book. Do you see any problems? $\endgroup$
    – Anton Klyachko
    Oct 11, 2012 at 13:41

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