## 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.].
Reviewer: Ioan Pop (Iaşi)

### MSC:

 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010) 57M05 Fundamental group, presentations, free differential calculus

### Citations:

Zbl 0849.53027; Zbl 0909.53018; Zbl 1031.57021; Zbl 1173.57013
Full Text:

### 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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