Valuations, trees, and degenerations of hyperbolic structures. I.

*(English)*Zbl 0583.57005The theory developed in this paper arose from two main sources. They are the theory of varieties of group representations developed recently by M. Culler and the second author [ibid. 117, 109-146 (1983; Zbl 0529.57005)] and Thurston’s construction of a compactification of Teichmüller space. As an application of their ideas, the authors give a new construction of this compactification. As they state, their ”methods are drawn from the mathematical mainstream, and therefore help to explain Thurston’s results by putting them in a wider framework.”

The central topic of the paper is a construction of compactifications of real and complex algebraic varieties. While there is an obvious way to compactify curves, which was used by Culler-Shalen, the problem of compactification of higher dimensional varieties is anything but routine. The authors’ approach to this problem is motivated by the construction of Thurston’s compactification of Teichmüller space. They consider an affine algebraic set V and an indexed family \((f_ j)_{j\in J}={\mathcal F}\) with countable index set J of functions which belong to the coordinate ring of V and generate it as an algebra. A compactification of V is canonically defined by \({\mathcal F}\) as follows. Let \({\mathcal P}\) be the quotient of \([0,\infty)^ J\setminus \{0\}\), where \([0,\infty)^ J\) is the Cartesian power, by the diagonal action of positive reals: \(\alpha (t_ j)_{j\in J}=(\alpha t_ j)_{j\in J}.\) Define a map \(\theta\) : \(V\to {\mathcal P}\) by \(\theta (x)=[\log (| f_ j(x)| +2)]_{j\in J}.\) Then the closure of \(\theta\) (V) in \({\mathcal P}\) is compact. This closure is the compactification in question. This compactification is studied on three different levels of generality.

The first one is that of a general variety. This is the theme of Chapter I. The points added to V are interpreted as valuations of the coordinate ring of V over a countable field of definition of V. These valuations are neither discrete nor of rank 1 in general. An important result says that there is a dense subset of added points consisting of discrete, rank 1 valuations.

At the second level V specializes to be the variety of characters X(\(\Gamma)\) of representations of a discrete group \(\Gamma\) in \(SL_ 2({\mathbb{C}})\). In this case there is a natural choice of \({\mathcal F}\). The corresponding \(f_ j\) are the values of characters on conjugacy classes in \(\Gamma\). Now the added points can be interpreted as actions of \(\Gamma\) on some generalized trees. On the vertices of an ordinary tree there is an integer-valued distance function. On generalized trees a similar distance function takes values in an ordered abelian group. The most important case is that of a subgroup of \({\mathbb{R}}\). The theory of such trees is developed from scratch up to a generalization of the well known Bass-Serre theory of trees associated to \(SL_ 2(F)\), where F is a field. While in the Bass-Serre theory a tree is associated with a discrete valuation of F, here the valuation can be nondiscrete. These trees are used for compactification. All this is the theme of Chapter II.

Finally, in Chapter III this theory is applied to the case \(\Gamma =\pi_ 1(S)\), where S is a surface. The Teichmüller space of S turns out to be a component of X(\(\Gamma)\) and the compactification of X(\(\Gamma)\) constructed in Chapter II leads to the Thurston’s compactification of Teichmüller space.

In the second part of this paper, existing now in preprint form, these ideas are applied to the study of 3-manifolds. In particular, another important result of Thurston is re-proved from an entirely new point of view.

The central topic of the paper is a construction of compactifications of real and complex algebraic varieties. While there is an obvious way to compactify curves, which was used by Culler-Shalen, the problem of compactification of higher dimensional varieties is anything but routine. The authors’ approach to this problem is motivated by the construction of Thurston’s compactification of Teichmüller space. They consider an affine algebraic set V and an indexed family \((f_ j)_{j\in J}={\mathcal F}\) with countable index set J of functions which belong to the coordinate ring of V and generate it as an algebra. A compactification of V is canonically defined by \({\mathcal F}\) as follows. Let \({\mathcal P}\) be the quotient of \([0,\infty)^ J\setminus \{0\}\), where \([0,\infty)^ J\) is the Cartesian power, by the diagonal action of positive reals: \(\alpha (t_ j)_{j\in J}=(\alpha t_ j)_{j\in J}.\) Define a map \(\theta\) : \(V\to {\mathcal P}\) by \(\theta (x)=[\log (| f_ j(x)| +2)]_{j\in J}.\) Then the closure of \(\theta\) (V) in \({\mathcal P}\) is compact. This closure is the compactification in question. This compactification is studied on three different levels of generality.

The first one is that of a general variety. This is the theme of Chapter I. The points added to V are interpreted as valuations of the coordinate ring of V over a countable field of definition of V. These valuations are neither discrete nor of rank 1 in general. An important result says that there is a dense subset of added points consisting of discrete, rank 1 valuations.

At the second level V specializes to be the variety of characters X(\(\Gamma)\) of representations of a discrete group \(\Gamma\) in \(SL_ 2({\mathbb{C}})\). In this case there is a natural choice of \({\mathcal F}\). The corresponding \(f_ j\) are the values of characters on conjugacy classes in \(\Gamma\). Now the added points can be interpreted as actions of \(\Gamma\) on some generalized trees. On the vertices of an ordinary tree there is an integer-valued distance function. On generalized trees a similar distance function takes values in an ordered abelian group. The most important case is that of a subgroup of \({\mathbb{R}}\). The theory of such trees is developed from scratch up to a generalization of the well known Bass-Serre theory of trees associated to \(SL_ 2(F)\), where F is a field. While in the Bass-Serre theory a tree is associated with a discrete valuation of F, here the valuation can be nondiscrete. These trees are used for compactification. All this is the theme of Chapter II.

Finally, in Chapter III this theory is applied to the case \(\Gamma =\pi_ 1(S)\), where S is a surface. The Teichmüller space of S turns out to be a component of X(\(\Gamma)\) and the compactification of X(\(\Gamma)\) constructed in Chapter II leads to the Thurston’s compactification of Teichmüller space.

In the second part of this paper, existing now in preprint form, these ideas are applied to the study of 3-manifolds. In particular, another important result of Thurston is re-proved from an entirely new point of view.

Reviewer: N.V.Ivanov

##### MSC:

57M99 | General low-dimensional topology |

14K99 | Abelian varieties and schemes |

13A18 | Valuations and their generalizations for commutative rings |

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |

32J05 | Compactification of analytic spaces |

20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |

30F20 | Classification theory of Riemann surfaces |

57N05 | Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) |

57N10 | Topology of general \(3\)-manifolds (MSC2010) |