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Stable presentation length of 3-manifold groups. (English) Zbl 1396.57002
The author studies stable presentation length of $$3$$-manifold groups. He proves the stable presentation length is additive under sphere and torus decompositions of $$3$$-manifolds, and the simplicial volume of a closed $$3$$-manifold $$M$$ is bounded from above and below by multiples of the stable presentation length of $$\pi_1(M)$$.
For a finitely presented group $$G$$, its presentation length is defined to be $T(G)=\min_{\mathcal{P}}(\sum_{i=1}^m\max{\{0,|r_i|-2\}}),$ where $$\mathcal{P}$$ runs over all presentations $$\mathcal{P}=\langle x_1,\ldots,x_n\;|\;r_1,\ldots,r_m\rangle$$ of $$G$$. The stable presentation length of $$G$$ is defined to be $T_{\infty}(G)=\inf_{H\leq G}\frac{T(H)}{[G:H]}$ where $$H$$ runs over all finite index subgroups of $$G$$. Note that $$T_{\infty}(G)$$ is a “volume-like” invariant of a group (or a $$3$$-manifold), in the sense that if $$H\leq G$$ is a finite-index subgroup, then we have $T_{\infty}(H)=[G:H]\cdot T_{\infty}(G).$
The main results in this paper are the following ones.
1. For finitely presented groups $$G_1$$ and $$G_2$$, we have $T_{\infty}(G_1*G_2)=T_{\infty}(G_1)+T_{\infty}(G_2).$ 2. Let $$M$$ be a compact irreducible $$3$$-manifold with empty or tori boundary, and let the JSJ decomposition of $$M$$ be $$M=M_1\cup M_2\cup \cdots\cup M_h$$, then $T_{\infty}(M)=T_{\infty}(M_1)+T_{\infty}(M_2)+\cdots+T_{\infty}(M_h).$ 3. There exists a constant $$C>0$$, such that for any closed $$3$$-manifold $$M$$, its stable presentation length and simplicial volume satisfies $C\cdot T_{\infty}(M)\leq \|M\|\leq \frac{\pi}{V_3}T_{\infty}(M).$
For the first result, its proof only uses standard arguments on presentation length. The proof of the second result invokes the relative stable presentation length $$T_{\infty}(G;C_1,C_2,\ldots,C_l)$$ of a group $$G$$ relative to its finitely many subgroups $$C_1,C_2,\ldots,C_l$$. In particular, the author proves that if all $$C_i$$ are free abelian groups of rank at least $$2$$, then $T_{\infty}(G;C_1,C_2,\ldots,C_l)=T_{\infty}(G)$ holds. The proof of the third result uses the relation between the simplicial volume and the stable $$\Delta$$-complexity of hyperbolic $$3$$-manifolds.
The author also computes the stable presentation length of surface groups, and gives estimations on stable presentation lengths of fundamental groups of various arithmetic link complements.
##### MSC:
 57M05 Fundamental group, presentations, free differential calculus 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 57M10 Covering spaces and low-dimensional topology 57M20 Two-dimensional complexes (manifolds) (MSC2010) 57N10 Topology of general $$3$$-manifolds (MSC2010)
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