Dear all,
I came across the above group (for any fixed odd prime $p$) and would need to know if it's large or not. It seems like it shouldn't be...
Best wishes, Elisabeth
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$\begingroup$ What do you mean by "small"? I assume that $p$ is a fixed natural number. The group is certainly infinite because $b_1,b_2,b_3$ generate a copy of $\mathcal{Z}^3$. Whenever $p>1$, $G$ is not amenable because $b_1,b_2b_1b_2^{-1}$ generates a free group $\endgroup$– Steven DeprezCommented Apr 26, 2012 at 15:20
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$\begingroup$ Maybe "large" in the sense of subjecting onto a free group? $\endgroup$– Igor RivinCommented Apr 26, 2012 at 15:25
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$\begingroup$ subjecting -> suRjecting. Thanks, Apple, for the spelling correction... $\endgroup$– Igor RivinCommented Apr 26, 2012 at 15:45
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1$\begingroup$ @Igor: large means there's a finite-index subgroup surjecting a rank 2 free group. $\endgroup$– Ian AgolCommented Apr 26, 2012 at 15:54
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1$\begingroup$ Your group is obtained by adjoining roots to each generator of $\mathbb{Z}^3$. In general, if you add an $n$th root to an element $\gamma$ of a group $\Gamma$, then there is an index-$n$ subgroup isomorphic to an iterated $n$-fold amalgamated product of $\Gamma$ along $\gamma$. This subgroup is the kernel of the map to $\mathbb{Z}/n$ that kills $\Gamma$. $\endgroup$– HJRWCommented Apr 29, 2012 at 19:24
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1 Answer
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The group $G$ maps onto the free product $C_p*C_p*C_p$ of three cyclic groups of order $p$ (just send each $b_i^p$ to $1$). This free product is virtually free, as a free product of finite groups (by Kurosh theorem the kernel of the homomorphism onto $C_p \times C_p \times C_p$ is free) and is not virtually cyclic. Hence it is large, and so $G$ is large as well. This works for any $p>1$.
