Lattices, i.e. discrete subgroups of finite covolume, in semisimple Lie groups, their structure and their classification form the central theme of this book. More generally, the notion of ‘Lie group’ may also include the sets of k-rational points of algebraic groups over local fields and their direct products. The results are applicable to the theory of algebraic groups defined over algebraic number fields.
A prominent class of examples of lattices is provided by the following construction: Let \(G\) be an algebraic subgroup of the general linear group \(\mathrm{GL}_ n\) defined over the field \({\mathbb Q}\) of rational numbers. Thus there exists a set of polynomials with rational coefficients in the matrix entries and the inverse of the determinant, whose set of solutions in any field extension \(F\) of \({\mathbb Q}\) is a subgroup \(G(F)\) of \(\mathrm{GL}_ n(F)\). The group \(G(F)\) is called the group of \(F\)-rational points of \(G\). Then the groups \(G({\mathbb R})\) and \(G({\mathbb C})\) are respectively real and complex Lie groups. A subgroup \(\Gamma\) of \(G({\mathbb Q})\) is called arithmetic if it is commensurable with \(\mathrm{GL}_ n({\mathbb Z})\cap G({\mathbb Q})_ 0\); viewed as a subgroup of G(\({\mathbb R})\Gamma\) is a discrete group. One may replace the field \({\mathbb Q}\) in the definition by an arbitrary number field \(F\), and \({\mathbb Z}\) by the ring of integers \({\mathcal O}_ F\). However, by use of “restriction of scalars”, this does not enlarge the class of “arithmetic groups”.
It is known that the group \(\mathrm{SL}_ 2({\mathbb R})\) contains nonarithmetic lattices. Most Fuchsian groups in \(\mathrm{SL}_ 2({\mathbb R})\) give examples. And the existence of nonarithmetic lattices is due to the fact that in \(\mathrm{SL}_ 2({\mathbb R})\) there are lattices which are not locally rigid. However, following the local rigidity theorem due to Selberg-Weil, one has for a given connected semisimple Lie group \(G\) without nontrivial compact factor groups and which is not locally isomorphic to \(\mathrm{SL}_ 2({\mathbb R})\) that any irreducible cocompact lattice \(\Gamma\subset G\) is locally rigid. If in addition \(G\) is not locally isomorphic to \(\mathrm{SL}_ 2({\mathbb C})\), then the analogous assertion is also valid for non-cocompact lattices. This had led to the conjecture that if \(G\) is a connected semisimple Lie group without nontrivial compact factor groups and which is not locally isomorphic to \(\mathrm{SL}_ 2({\mathbb R})\) then any irreducible lattice \(\Gamma\subset G\) is arithmetic, i.e. there exists a \({\mathbb Q}\)-algebraic group \(H\) and a Lie group homomorphism \(\phi : H({\mathbb R})\to G\) such that \(\phi\) has compact kernel and open image and \(\phi(\Lambda\)) is commensurable with \(\Gamma\) for any arithmetic subgroup \(\Lambda\) of \(G({\mathbb Q})\). Then in the mid 70’s Margulis has proved an assertion stating that the conjecture is ‘essentially’ true, namely in case \(\mathrm{rank}_{{\mathbb R}}G>1\). This arithmeticity result, or better its most general form, forms the core of the book under review.
On the way towards its complete proof there are, besides the methods of algebraic group theory, the techniques in ergodic theory, measure theory or the theory of infinite-dimensional unitary representations of great help. All this material is quite carefully developed and presented in a very clear way by the author, being not afraid of any technical detail.
The book is divided into nine chapters. Following Chapter I recalling as preliminaries mostly the main results in algebraic group theory Chapter II deals mainly with density theorems, in particular, with a version of the Borel-Wang theorem on Zariski density of certain subgroups in algebraic groups.
In Chapter III a discussion of the notion ‘property (T)’ as suggested by Kazhdan, is given. For a locally compact group this means that the trivial 1-dimensional representation is isolated in the space of irreducible unitary representations. For discrete groups property (T) is closely connected with their structure. Lattices in a broad class of semisimple groups have this property (T). As a consequence, such a lattice \(\Gamma\) is finitely generated and the factor group of the group modulo its commutator subgroup is finite. These results are combined in Chapter IV with a study of invariant algebras of measurable sets to obtain finiteness theorems for factor groups of discrete groups. Having dealt with questions of rigidity this theme is taken up in a more general form again in Chapter VIII.
The headings of chapter V and VI are ‘Characteristic maps’ and ‘Discrete subgroups and boundary theory’. The theorems on existence of equivariant measurable maps proved there are used later on in dealing with questions of superrigidity. The approach in chapter V is based on the multiplicative ergodic theorem, whereas the Furstenberg boundary theory is fundamental for the approach presented in chapter VI. It contains a comprehensive introduction into this theory.
In Chapter VII the superrigidity theorems for discrete subgroups are established, i.e. results concerning continuous extensions of homomorphisms of discrete subgroups to algebraic groups over local fields. The proofs are based on the study of equivariant measurable maps to linear spaces. To indicate what these results are about we indicate its assertion in a Lie theoretic formulation in a very special case: Let \(H\) be a connected semisimple Lie group without nontrivial center, let \(\Gamma\subset H\) be a lattice, \(\Lambda\supset \Gamma\) a subgroup of the commensurability subgroup \(C(\Gamma)=\{h\in H\mid\) the subgroups \(h\Gamma h^{-1}\) and \(\Gamma\) are commensurable\(\}\), \(F\) a connected semisimple \({\mathbb R}\)-group which is adjoint and has no nontrivial \({\mathbb R}\)-anisotropic factors, and let \(\gamma: \Lambda\to F({\mathbb R})\) be a homomorphism such that the subgroup \(\gamma\) (\(\Gamma\)) is Zariski dense in \(F\). Suppose that either \(\text{rank}\;H\geq 2\) and the lattice \(\Gamma\) is irreducible or the subgroup \(\Lambda\) is dense in H. Then the homomorphism \(\gamma\) extends uniquely to a continuous homomorphism \({\tilde \gamma}: H\to F({\mathbb R})\). In particular, one has \({\tilde \gamma}(H)=F({\mathbb R})^ 0.\)
Some applications of the superrigidity theorems and the finiteness theorems for factor groups of discrete subgroups to the study of algebraic groups over global fields are given in Chapter VIII. Finiteness of factor groups of S-arithmetic subgroups and homomorphisms of S- arithmetic subgroups to algebraic groups are the main topics.
In Chapter IX the arithmeticity theorems are given in full generality and a series of consequences of these theorems are discussed, in particular, certain previously proved results concerning finiteness of factor groups and rigidity respectively can be strengthened. In the final sections the previous results are restated in terms of Lie group theory, the theory of symmetric spaces and the theory of complex manifolds. Among these applications one finds e.g. a strengthening of Mostow’s theorem on the strong ridigity of locally symmetric spaces, for spaces of rank \(\geq 2.\)
The main body of the book is supplemented by three appendices, two of them of a technical nature, the other one providing some examples of nonarithmetic lattices in real rank 1 semisimple Lie groups.
The book contains extensive references and some historical and bibliographical notes. From the list of references there are two books which should be mentioned. In his book ‘Discrete subgroups of Lie groups’ (Berlin etc.: Springer) which appeared in 1972 (Zbl 0254.22005) M. S. Raghunathan gave a detailed treatment of the more geometric aspects of discrete groups. In particular, he covered the theory of lattices in nilpotent and solvable Lie groups, gave proofs of Borel’s density theorem and the local rigidity theorem of Selberg-Weil. His treatment of discrete subgroups of semisimple Lie groups is naturally limited by the year of publication of his book, i.e. omitting important developments. But these are the core of the book under review. Margulis’ work on rigidity, arithmeticity, and normal subgroups of lattices in semisimple groups is treated as well in R. J. Zimmer’s book ‘Ergodic theory and semisimple groups’ (Boston etc.: Birkhäuser) which appeared in 1984 (Zbl 0571.58015). It made the subject accessible to mathematicians with little background in ergodic theory or algebraic group theory but is not as comprehensive and general in its outlet as Margulis’ book. Surely the latter one will serve as a reference point.
The book of Margulis does what it says it will do, namely to give a coherent presentation of the theory of lattices, their structure and their classification. It is a demanding but rewarding book on a subject which has been extensively explored over the past forty years. The patient reader will come away with a real sense of the accomplishments of these explorations.