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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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