Subgroups of Teichmüller modular groups. Translated from the Russian by E. J. F. Primrose.

*(English)*Zbl 0776.57001
Translations of Mathematical Monographs. 115. Providence, RI: American Mathematical Society (AMS). xii, 127 p. (1992).

After linear groups and their discrete subgroups (most notably \(\text{SL}(2,\mathbb{Z})\) and the classical modular group \(\text{PSL}(2,\mathbb{Z}))\), some of the most interesting infinite groups, with extremely rich structure, are the modular groups or mapping class groups \(\text{Mod}_ S\) of compact orientable surfaces \(S\). These groups occur in various fields of mathematics: in topology as the group of isotopy classes of diffeomorphisms of \(S\) (mapping class group), in group theory as the outer automorphism group of the fundamental group of \(S\), and in Riemann surfaces and algebraic geometry as the Teichmüller modular group acting on the Teichmüller space of \(S\). This action of the modular group \(\text{Mod}_ S\) on Teichmüller space extends to the Thurston compactification of Teichmüller space by the projective space \(\text{PF}_ S\) of measured foliations on \(S\), and the action of \(\text{Mod}_ S\) on the compact space \(\text{PF}_ S\) gives a powerful tool to study modular groups, in a similar way as linear groups can be studied geometrically by their actions on linear and projective spaces. Using this action, Thurston gave a classification of elements of \(\text{Mod}_ S\) into 3 types: finite order, reducible (leaving invariant, up to isotopy, a system of disjointly embedded essential closed curves on \(S\)) and pseudo-Anosov (or irreducible), rediscovering and lifting onto a broader basis old and rather unnoticed work of Nielsen. The main point of the present work is the study of arbitrary subgroups of \(\text{Mod}_ S\) in a similar way, i.e. by geometric methods using the action on \(\text{PF}_ S\), where well-known theorems about linear groups obtained by their actions on projective spaces serve as a guiding principle. Again, there is an obvious notion of reducible and irreducible subgroups of \(\text{Mod}_ S\) (the finite case being subject of much interest already, most notably in the Nielsen realization problem). Some of the main results are:

Theorem. An infinite irreducible subgroup of a modular group either contains, as a subgroup of finite index, an infinite cyclic subgroup generated by a pseudo-Anosov element or contains two pseudo-Anosov elements that generate a free group.

Theorem. Each subgroup of a modular group either contains a free subgroup wtih 2 generators, or it is almost (virtually) Abelian.

The last one corresponds to the “Tits alternative” for linear groups (with Abelian replaced by solvable). The proofs require an efficient technical description of the action of \(\text{Mod}_ S\) on \(\text{PF}_ S\) and, in the reducible case, good decomposition techniques: here lies the main work of the book. The notion of a pure diffeomorphism (or mapping class) is introduced: one which decomposes along an essential curve system on \(S\) such that, after cutting, it preserves each component where it is (isotopic to) the identity or pseudo-Anosov. It is shown that the kernel \(\Gamma_ m\) of the canonical map \(\text{Mod}_ S\to\operatorname{Aut}(H_ 1(S,\mathbb{Z}/m\mathbb{Z}))\) consists of pure diffeomorphisms, for \(m>2\) (implying and generalizing the result that \(\text{Mod}_ S\) has a torsion-free subgroup of finite index). Then, for a subgroup \(G\) of \(\text{Mod}_ S\), one may consider, \(G\cap\Gamma_ m\), first the irreducible case and then, using the notion of a canonical reduction system, the reducible case. So, in a sense, the basic case are groups consisting purely of pseudo-Anosov mapping classes. Here freeness of certain subgroups can be proved by a basic geometric method already used by Klein, Poincaré and Schottky in the case of Möbius transformations, where in the present situation the stable and unstable foliations of pseudo-Anosovs correspond to the attracting and repelling fixed points of loxodromic transformations. Here lies the basic geometric idea, but, as noted above, the main work lies in developing efficient techniques which make these concepts work.

Other analogues of theorems about linear groups proved along these lines are as follows. Every finitely generated subgroup \(G\) of a modular group either is almost Abelian, or contains a maximal subgroup of infinite index, in fact uncountably many such subgroups. Also, the Frattini subgroup of \(G\) (the intersection of all maximal subgroups) is nilpotent. So there exists a strong analogy between linear and modular groups. However, \(\text{Mod}_ S\) itself does not seem to be linear, see the remarks (Section 5) in a paper by J. L. Harer [Invent. Math. 84, 157-176 (1986; Zbl 0592.57009)]; this paper gives also an approach to another important recent direction of research on modular groups, its cohomological properties which are not mentioned in the present work.

As noted in the introduction of the book, a stimulus for the present work came from a paper of J. S. Birman, A. Lubotzky and J. McCarthy [Duke Math. J. 50, 1107-1120 (1983; Zbl 0551.57004)]. This was continued (independently of the present work) by McCarthy who also proved the Tits alternative for modular groups [Trans. Am. Math. Soc. 291, 583- 612 (1985; Zbl 0579.57006)]. The basic ideas are similar but the technical details differ, and the present work goes further. To conclude, this is an impressive and well-written book on persistent work of the author’s over the last decade. It is both interesting for its results as well as for its methods and gives an excellent opportunity to see Thurston’s theory of surface diffeomorphisms at work.

Theorem. An infinite irreducible subgroup of a modular group either contains, as a subgroup of finite index, an infinite cyclic subgroup generated by a pseudo-Anosov element or contains two pseudo-Anosov elements that generate a free group.

Theorem. Each subgroup of a modular group either contains a free subgroup wtih 2 generators, or it is almost (virtually) Abelian.

The last one corresponds to the “Tits alternative” for linear groups (with Abelian replaced by solvable). The proofs require an efficient technical description of the action of \(\text{Mod}_ S\) on \(\text{PF}_ S\) and, in the reducible case, good decomposition techniques: here lies the main work of the book. The notion of a pure diffeomorphism (or mapping class) is introduced: one which decomposes along an essential curve system on \(S\) such that, after cutting, it preserves each component where it is (isotopic to) the identity or pseudo-Anosov. It is shown that the kernel \(\Gamma_ m\) of the canonical map \(\text{Mod}_ S\to\operatorname{Aut}(H_ 1(S,\mathbb{Z}/m\mathbb{Z}))\) consists of pure diffeomorphisms, for \(m>2\) (implying and generalizing the result that \(\text{Mod}_ S\) has a torsion-free subgroup of finite index). Then, for a subgroup \(G\) of \(\text{Mod}_ S\), one may consider, \(G\cap\Gamma_ m\), first the irreducible case and then, using the notion of a canonical reduction system, the reducible case. So, in a sense, the basic case are groups consisting purely of pseudo-Anosov mapping classes. Here freeness of certain subgroups can be proved by a basic geometric method already used by Klein, Poincaré and Schottky in the case of Möbius transformations, where in the present situation the stable and unstable foliations of pseudo-Anosovs correspond to the attracting and repelling fixed points of loxodromic transformations. Here lies the basic geometric idea, but, as noted above, the main work lies in developing efficient techniques which make these concepts work.

Other analogues of theorems about linear groups proved along these lines are as follows. Every finitely generated subgroup \(G\) of a modular group either is almost Abelian, or contains a maximal subgroup of infinite index, in fact uncountably many such subgroups. Also, the Frattini subgroup of \(G\) (the intersection of all maximal subgroups) is nilpotent. So there exists a strong analogy between linear and modular groups. However, \(\text{Mod}_ S\) itself does not seem to be linear, see the remarks (Section 5) in a paper by J. L. Harer [Invent. Math. 84, 157-176 (1986; Zbl 0592.57009)]; this paper gives also an approach to another important recent direction of research on modular groups, its cohomological properties which are not mentioned in the present work.

As noted in the introduction of the book, a stimulus for the present work came from a paper of J. S. Birman, A. Lubotzky and J. McCarthy [Duke Math. J. 50, 1107-1120 (1983; Zbl 0551.57004)]. This was continued (independently of the present work) by McCarthy who also proved the Tits alternative for modular groups [Trans. Am. Math. Soc. 291, 583- 612 (1985; Zbl 0579.57006)]. The basic ideas are similar but the technical details differ, and the present work goes further. To conclude, this is an impressive and well-written book on persistent work of the author’s over the last decade. It is both interesting for its results as well as for its methods and gives an excellent opportunity to see Thurston’s theory of surface diffeomorphisms at work.

Reviewer: B.Zimmermann (Trieste)

##### MSC:

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |

57M99 | General low-dimensional topology |

20H05 | Unimodular groups, congruence subgroups (group-theoretic aspects) |

30F60 | Teichmüller theory for Riemann surfaces |

20F65 | Geometric group theory |