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**On genera of Lefschetz fibrations and finitely presented groups.**
*(English)*
Zbl 1343.57010

Based on some results of R. E. Gompf [Ann. Math. (2) 142, No. 3, 527–595 (1995; Zbl 0849.53027)] and S. K. Donaldson [Doc. Math., J. DMV , 309–314 (1998; Zbl 0909.53018)] it follows that every finitely presented group is the fundamental group of the total space of a Lefschetz fibration. Then J. Amorós et al. [J. Differ. Geom. 54, No. 3, 489–545 (2000; Zbl 1031.57021)] and M. Korkmaz [Int. Math. Res. Not. 2009, No. 9, 1547–1572 (2009; Zbl 1173.57013)] constructed Lefschetz fibrations whose fundamental groups are a given finitely presented group. In particular, Korkmaz [loc. cit.] provided explicitely genus and monodromy of such a Lefschetz fibration.

Let \(F_n=\langle g_1,\dots,g_n\rangle \) be the free group of rank \(n\). For \(x\in F_n\), the syllable length \(l(x)\) of \(x\) is defined by \[ l(x)=\min \{s\mid x=g^{m(1)}_{i(1)}\dots g^{m(s)}_{i(s)}, 1\leq i(j)\leq n, m(j)\in \mathbb{Z}\}. \] For a finitely presented group \(\Gamma\) with a presentation \(\Gamma=\langle g_ 1,\dots,g_n\mid r_1,\dots,r_k\}\), Korkmaz [loc. cit.] proved that for any \[ g\geq 2(n+\sum_{1\leq i\leq k}l(r_i)-k) \] there exists a genus-\(g\) Lefschetz fibration \(f:X\rightarrow S^2\) such that the fundamental group \(\pi_1(X)\) is isomorphic to \(\Gamma\), proving explicitely a monodromy.

In the present paper the author proves the following theorem.

Theorem 1.1. Let \(\Gamma\) be a finitely presented group with a presentation \(\Gamma=\langle g_1,\dots,g_n\mid r_1,\dots,r_k\rangle \), and let \(l=\max_{1\leq i\leq k}\{l(r_i)\}\). Then for any \(g\geq 2n+l-1\), there exists a genus-\(g\) Lefschetz fibration \(f:X\rightarrow S^2\) such that the fundamental group \(\pi_1(X)\) is isomorphic to \(\Gamma\). For \(k=0\) is taken \(l=1\).

Since \(2(n+\sum_{1\leq i\leq k}l(r_i)-k)\geq 2n+l-1\), Theorem 1.1 is an improvement of Korkmaz’ result.

In addition, Korkmaz [loc. cit.] defined the genus \(g(\Gamma)\) of a finitely presented group \(\Gamma\) to be the minimal genus of a Lefschetz fibration with sections whose fundamental group is isomorphic to \(\Gamma\). But the Lefschetz fibrations which are constructed in the proof of Theorem 1.1 have sections, such that the genus of a finitely presented group is always well-defined.

A second important theorem of the paper evaluates upper bounds for genera of some finitely presented groups.

Theorem 1.2. (1) Let \(B_n\) denote the \(n\)-strand braid group. Then for \(n\geq 3\), we have \(2\leq g(B_n)\leq 4 \).

(2) Let \(\mathcal{H}_g\) be the hyperelliptic mapping class group of a closed connected orientable surface of genus \(g\geq 1\). Then we have \(2\leq g(\mathcal{H}_g)\leq 4\).

(3) Let \(\mathcal{M}_{0,n}\) denote the mapping class group of a sphere with \(n\) punctures. Then for \(n\geq 3\), we have \(2\leq g(\mathcal{M}_{0,n})\leq 4\).

(4) Let \(S_n\) denote the \(n\)-symmetric group. Then for \(n\geq 3\), we have \(2\leq g(S_n)\leq 4\).

(5) Let \(\mathcal{A}_n\) denote the \(n\)-Artin group associated to the Dynkin diagram shown in Fig.1. Then for \(n\geq 6\), we have \(2\leq g(\mathcal{A}_n)\leq 5\).

(6) Let \(n,k\geq 0\) be integers with \(n+k\geq 3\), and let \(m_1,\dots, m_k\geq 2\) be integers. Then we have \((n+k+1)/2\leq g(\mathbb{Z}^n\oplus \mathbb{Z}_{m_1}\oplus\dots\oplus \mathbb{Z}_{m_k})\leq n+k+1\).

The inequalities (1) and (6) improve some results of Korkmaz [loc. cit.].

Let \(F_n=\langle g_1,\dots,g_n\rangle \) be the free group of rank \(n\). For \(x\in F_n\), the syllable length \(l(x)\) of \(x\) is defined by \[ l(x)=\min \{s\mid x=g^{m(1)}_{i(1)}\dots g^{m(s)}_{i(s)}, 1\leq i(j)\leq n, m(j)\in \mathbb{Z}\}. \] For a finitely presented group \(\Gamma\) with a presentation \(\Gamma=\langle g_ 1,\dots,g_n\mid r_1,\dots,r_k\}\), Korkmaz [loc. cit.] proved that for any \[ g\geq 2(n+\sum_{1\leq i\leq k}l(r_i)-k) \] there exists a genus-\(g\) Lefschetz fibration \(f:X\rightarrow S^2\) such that the fundamental group \(\pi_1(X)\) is isomorphic to \(\Gamma\), proving explicitely a monodromy.

In the present paper the author proves the following theorem.

Theorem 1.1. Let \(\Gamma\) be a finitely presented group with a presentation \(\Gamma=\langle g_1,\dots,g_n\mid r_1,\dots,r_k\rangle \), and let \(l=\max_{1\leq i\leq k}\{l(r_i)\}\). Then for any \(g\geq 2n+l-1\), there exists a genus-\(g\) Lefschetz fibration \(f:X\rightarrow S^2\) such that the fundamental group \(\pi_1(X)\) is isomorphic to \(\Gamma\). For \(k=0\) is taken \(l=1\).

Since \(2(n+\sum_{1\leq i\leq k}l(r_i)-k)\geq 2n+l-1\), Theorem 1.1 is an improvement of Korkmaz’ result.

In addition, Korkmaz [loc. cit.] defined the genus \(g(\Gamma)\) of a finitely presented group \(\Gamma\) to be the minimal genus of a Lefschetz fibration with sections whose fundamental group is isomorphic to \(\Gamma\). But the Lefschetz fibrations which are constructed in the proof of Theorem 1.1 have sections, such that the genus of a finitely presented group is always well-defined.

A second important theorem of the paper evaluates upper bounds for genera of some finitely presented groups.

Theorem 1.2. (1) Let \(B_n\) denote the \(n\)-strand braid group. Then for \(n\geq 3\), we have \(2\leq g(B_n)\leq 4 \).

(2) Let \(\mathcal{H}_g\) be the hyperelliptic mapping class group of a closed connected orientable surface of genus \(g\geq 1\). Then we have \(2\leq g(\mathcal{H}_g)\leq 4\).

(3) Let \(\mathcal{M}_{0,n}\) denote the mapping class group of a sphere with \(n\) punctures. Then for \(n\geq 3\), we have \(2\leq g(\mathcal{M}_{0,n})\leq 4\).

(4) Let \(S_n\) denote the \(n\)-symmetric group. Then for \(n\geq 3\), we have \(2\leq g(S_n)\leq 4\).

(5) Let \(\mathcal{A}_n\) denote the \(n\)-Artin group associated to the Dynkin diagram shown in Fig.1. Then for \(n\geq 6\), we have \(2\leq g(\mathcal{A}_n)\leq 5\).

(6) Let \(n,k\geq 0\) be integers with \(n+k\geq 3\), and let \(m_1,\dots, m_k\geq 2\) be integers. Then we have \((n+k+1)/2\leq g(\mathbb{Z}^n\oplus \mathbb{Z}_{m_1}\oplus\dots\oplus \mathbb{Z}_{m_k})\leq n+k+1\).

The inequalities (1) and (6) improve some results of Korkmaz [loc. cit.].

Reviewer: Ioan Pop (Iaşi)

### MSC:

57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |

57M05 | Fundamental group, presentations, free differential calculus |

### Keywords:

genus-\(g\) Lefschetz fibrations; fundamental group; genus of a finitely presented group; \(n\)-strand braid group; \(n\)-Artin group
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\textit{R. Kobayashi}, Osaka J. Math. 53, No. 2, 351--376 (2016; Zbl 1343.57010)

### References:

[1] | J. Amorós, F. Bogomolov, L. Katzarkov, T. Pantev: Symplectic Lefschetz fibrations with arbitrary fundamental groups , J. Differential Geom. 54 (2000), 489-545. · Zbl 1031.57021 |

[2] | J.S. Birman and H.M. Hilden: On the mapping class groups of closed surfaces as covering spaces ; in Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969), Ann. of Math. Studies 66 , Princeton Univ. Press, Princeton, NJ, 1971, 81-115. |

[3] | E. Brieskorn: Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe , Invent. Math. 12 (1971), 57-61. · Zbl 0204.56502 |

[4] | S.K. Donaldson: Lefschetz fibrations in symplectic geometry , Doc. Math. 1998 , 309-314. · Zbl 0909.53018 |

[5] | R.E. Gompf: A new construction of symplectic manifolds , Ann. of Math. (2) 142 (1995), 527-595. · Zbl 0849.53027 |

[6] | R.E. Gompf and A.I. Stipsicz: \(4\)-Manifolds and Kirby Calculus, Graduate Studies in Mathematics 20 , Amer. Math. Soc., Providence, RI, 1999. |

[7] | M. Korkmaz: Noncomplex smooth 4-manifolds with Lefschetz fibrations , Internat. Math. Res. Notices (2001), 115-128. · Zbl 0977.57020 |

[8] | M. Korkmaz: Lefschetz fibrations and an invariant of finitely presented groups , Internat. Math. Res. Notices (2009), 1547-1572. · Zbl 1173.57013 |

[9] | W. Magnus: Über Automorphismen von Fundamentalgruppen berandeter Flächen , Math. Ann. 109 (1934), 617-646. · Zbl 0009.03901 |

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