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In 1965, Murasugi [1] conjectured that any finitely presented group with deficiency at least two has trivial centre. The year before, he had proved it true for one-relator groups, and in [1] he proved it for non-abelian link groups.

Has there been any recent progress on the conjecture? Is it still open?

I suspect that the conjecture is still open, but being only vaguely familiar with this research I thought I'd make sure. For reference, some recent progress in the pro-$p$ setting seems to given in [2], where the conjecture is proved for pro-$p$ groups.

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[1] Murasugi, Kunio, On the center of the group of a link, Proc. Am. Math. Soc. 16, 1052-1057 (1965). ZBL0132.26704.

[2] Hillman, Jonathan A.; Schmidt, Alexander, Pro-(p) groups of positive deficiency., Bull. Lond. Math. Soc. 40, No. 6, 1065-1069 (2008). ZBL1162.20019.

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  • $\begingroup$ "For 1-relator groups": do you mean, for groups which are both 1-relator and deficiency $\ge 2$ (possibly for another presentation), or for groups with a 1-relator presentation of deficiency $\ge 2$? $\endgroup$
    – YCor
    Commented Oct 3 at 16:25
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    $\begingroup$ @YCor They are the same thing. The deficiency of a one-relator group with presentation $\langle a_1, \dots, a_n \mid R = 1 \rangle$ where $R$ is non-trivial is $n-1$, as proved by Rapaport. So Murasugi's result for one-relator groups is for all one-relator groups with $n \geq 3$. $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Oct 4 at 3:32

2 Answers 2

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All $L^2$-Betti numbers of a finitely generated group $G$ with an infinite amenable normal subgroup are 0, by a result of Gromov. (See Theorem 7.2 of Lück's book $L^2$-Invariants: Theory and Applications to Geometry and $K$-Theory.) Such a group has deficiency $\leq1$, with equality only if $G\cong\mathbb{Z}$ or $c.d.G=2$. (See "$L^2$-homology and asphericity", Israel J. M. 99 (1997), 271--283, by Hillman.) Thus if $G$ has deficiency 2 its centre must be finite.

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A group with deficiency at least 2 certainly has torsion centre so the conjecture holds for torsion-free groups. This is because it has positive first $\ell^2$-Betti number which implies it cannot have a finitely generated normal subgroup that is infinite and infinite index (I believe this is by work of Lück and Gaboriau), which any infinite cyclic subgroup of the centre would be.

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