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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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##### References:
  ; Agol, Doc. Math., 18, 1045, (2013) · Zbl 1286.57019  10.1090/S0894-0347-2011-00711-X · Zbl 1237.57002  10.5802/afst.1436 · Zbl 1322.57015  10.1017/S1474748012000667 · Zbl 1266.22013  10.1007/BF01388469 · Zbl 0538.57004  10.2140/pjm.2009.241.215 · Zbl 1228.30025  10.1007/978-3-642-62035-5  10.1112/jlms/jdw062 · Zbl 1358.57013  10.1090/S0002-9939-99-05190-4 · Zbl 1058.30037  10.1515/crll.1996.470.153 · Zbl 0836.20038  10.1007/s11856-012-0068-2 · Zbl 1271.57055  10.1112/jtopol/jts026 · Zbl 1266.57015  10.1007/s11856-016-1425-3 · Zbl 1361.57022  10.1112/S0024611599001823 · Zbl 1024.57020  ; Gromov, Inst. Hautes Études Sci. Publ. Math., 56, 5, (1982)  ; Haefliger, Group theory from a geometrical viewpoint, 504, (1991)  10.1112/jlms/s2-27.2.345 · Zbl 0516.57001  ; Hempel, Combinatorial group theory and topology. Ann. of Math. Stud., 111, 379, (1987)  10.4007/annals.2015.182.1.1 · Zbl 1400.57021  10.1090/conm/560/11089 · Zbl 1335.57028  10.1007/BF02773541 · Zbl 1066.22008  10.1007/s00222-005-0480-x · Zbl 1110.57015  10.1007/BF01457454 · Zbl 0488.12001  10.1090/S0894-0347-2013-00767-5 · Zbl 1277.57004  10.4171/CMH/169 · Zbl 1180.53046  10.1007/s00039-013-0218-7 · Zbl 1273.22009  10.4310/CAG.2006.v14.n5.a6 · Zbl 1118.57018  ; Matveev, Acta Appl. Math., 19, 101, (1990)  10.1090/conm/020/718149 · Zbl 0524.57005  ; Milnor, Enseignement Math., 23, 249, (1977)  ; Mostow, Strong rigidity of locally symmetric spaces. Annals of Mathematics Studies, 78, (1973) · Zbl 0265.53039  ; Neumann, Topology ’90. Ohio State Univ. Math. Res. Inst. Publ., 1, 273, (1992)  10.1002/mana.200510669 · Zbl 1193.20037  10.1007/BF01418789 · Zbl 0264.22009  10.1007/BF01445265 · Zbl 0859.20027  10.2140/agt.2015.15.2949 · Zbl 1346.57014  10.1007/978-3-642-61856-7  10.1007/BF01389276 · Zbl 0478.57006  10.2307/1970594 · Zbl 0157.30603
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