Trees. Transl. from the French by John Stillwell.
Corrected 2nd printing of the 1980 original.

*(English)*Zbl 1013.20001
Springer Monographs in Mathematics. Berlin: Springer. ix, 142 p. (2003).

The book under review is the corrected second printing of the English edition of 1980, which was a translation of the French edition published under the title Arbres, amalgams, \(\text{SL}_2\) [Astérisque 46, Soc. Math. France (1977; Zbl 0369.20013)], and written with the collaboration of H. Bass. The work was based on a part of a course given by the author at the Collège de France in 1968/69. The seminal ideas of the book played an important role in the development of group theory since the seventies; the Serre-Bass theory presented in the book was one of the sources of geometric group theory emerged within the last three decades. Nowdays the book already can be called classical.

The book consists of two chapters. The first chapter deals with the following question: what can be said of a group acting on a tree when we know the quotient graph under the action as well as the stabilizers of all vertices and edges? The author starts with the special case where the group acts freely; in this case the group is free. Then he considers another special case where the quotient graph is a segment, in which case the group can be identified with an amalgam of two groups; moreover, every amalgam can be, essentially uniquely, obtained in this way. Thus there is an equivalence between groups acting on trees with a segment as a fundamental domain, and amalgams. In the general case there is an equivalence between groups acting on trees such that the quotient is a tree, and direct limits of trees of groups (which are called tree products nowdays). The main result says, roughly, that one can reconstruct a group acting on a tree from the quotient tree and the stabilizers of vertices and edges; it is the ‘fundamental group’ of a ‘tree of groups’ carried by the quotient; conversely, every tree of groups can be, essentially uniquely, obtained in this way. Like in the case of amalgams, one can give a normal form for the elements of the direct limit of a tree of groups. The case where the quotient graph is a loop leads to HNN groups. As applications of groups acting on trees the author gives geometric proofs of Schreier’s theorem on subgroups of free groups and Kurosh’s theorem on subgroups of free products with amalgamated subgroup. The groups, any action of which on any tree has a fixed point, are studied; examples of groups with this property are \(\text{SL}_3(\mathbb{Z})\) and \(\text{Sp}_4(\mathbb{Z})\). It is shown that such a group cannot be an amalgam.

The second chapter gives applications to the group \(\text{SL}_2\) over a local field \(K\) and its action on a certain tree, which is a special case of a Bruhat-Tits building. For a 2-dimensional vector space \(V\) over \(K\), the vertices of this tree are the classes of lattices of \(V\), where two lattices are in the same class if they are homothetic; two vertices are adjacent if they are represented by nested lattices whose quotient is of length one. Applying the results of Chapter I, the author proves that \(\text{SL}_2(K)\) is an amalgam of two copies of \(\text{SL}_2(\mathcal O)\), where \(\mathcal O\) is the valuation ring of \(K\), and that every torsion-free discrete subgroup of \(\text{SL}_2(K)\) is free – Y. Ihara’s results which served as the starting point for the whole work. Besides, he proves the theorem of H. Nagao giving, for a commutative field \(k\), the structure of the group \(\text{GL}_2(k[t])\) as an amalgam of \(\text{GL}_2(k)\) and the Borel subgroup \(B(k[t])\). The latter result is generalized to the situation where \(k[t]\) is replaced by the affine algebra of a curve \(C^{\text{aff}}=C-\{P\}\) with a single point \(P\) at infinity.

Sometimes the proofs of results are only sketched. The book is full of remarks which show relations of the results with other parts of algebra and topology, and contains a lot of exercises (often difficult). Several generations of mathematicians learned geometric ideas in group theory from this stimulating book; without doubt, the new edition will be useful for graduate students and researchers working in algebra, geometry, and topology.

The book consists of two chapters. The first chapter deals with the following question: what can be said of a group acting on a tree when we know the quotient graph under the action as well as the stabilizers of all vertices and edges? The author starts with the special case where the group acts freely; in this case the group is free. Then he considers another special case where the quotient graph is a segment, in which case the group can be identified with an amalgam of two groups; moreover, every amalgam can be, essentially uniquely, obtained in this way. Thus there is an equivalence between groups acting on trees with a segment as a fundamental domain, and amalgams. In the general case there is an equivalence between groups acting on trees such that the quotient is a tree, and direct limits of trees of groups (which are called tree products nowdays). The main result says, roughly, that one can reconstruct a group acting on a tree from the quotient tree and the stabilizers of vertices and edges; it is the ‘fundamental group’ of a ‘tree of groups’ carried by the quotient; conversely, every tree of groups can be, essentially uniquely, obtained in this way. Like in the case of amalgams, one can give a normal form for the elements of the direct limit of a tree of groups. The case where the quotient graph is a loop leads to HNN groups. As applications of groups acting on trees the author gives geometric proofs of Schreier’s theorem on subgroups of free groups and Kurosh’s theorem on subgroups of free products with amalgamated subgroup. The groups, any action of which on any tree has a fixed point, are studied; examples of groups with this property are \(\text{SL}_3(\mathbb{Z})\) and \(\text{Sp}_4(\mathbb{Z})\). It is shown that such a group cannot be an amalgam.

The second chapter gives applications to the group \(\text{SL}_2\) over a local field \(K\) and its action on a certain tree, which is a special case of a Bruhat-Tits building. For a 2-dimensional vector space \(V\) over \(K\), the vertices of this tree are the classes of lattices of \(V\), where two lattices are in the same class if they are homothetic; two vertices are adjacent if they are represented by nested lattices whose quotient is of length one. Applying the results of Chapter I, the author proves that \(\text{SL}_2(K)\) is an amalgam of two copies of \(\text{SL}_2(\mathcal O)\), where \(\mathcal O\) is the valuation ring of \(K\), and that every torsion-free discrete subgroup of \(\text{SL}_2(K)\) is free – Y. Ihara’s results which served as the starting point for the whole work. Besides, he proves the theorem of H. Nagao giving, for a commutative field \(k\), the structure of the group \(\text{GL}_2(k[t])\) as an amalgam of \(\text{GL}_2(k)\) and the Borel subgroup \(B(k[t])\). The latter result is generalized to the situation where \(k[t]\) is replaced by the affine algebra of a curve \(C^{\text{aff}}=C-\{P\}\) with a single point \(P\) at infinity.

Sometimes the proofs of results are only sketched. The book is full of remarks which show relations of the results with other parts of algebra and topology, and contains a lot of exercises (often difficult). Several generations of mathematicians learned geometric ideas in group theory from this stimulating book; without doubt, the new edition will be useful for graduate students and researchers working in algebra, geometry, and topology.

Reviewer: Oleg V.Belegradek (İstanbul)

##### MSC:

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

20E08 | Groups acting on trees |

20G25 | Linear algebraic groups over local fields and their integers |

20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |

20F65 | Geometric group theory |

20E42 | Groups with a \(BN\)-pair; buildings |

05C05 | Trees |