Finite presentability of S-arithmetic groups. Compact presentability of solvable groups
This monograph gives a self-contained proof of what is essentially a single result (with a considerable number of consequences). The proof has not appeared elswhere in the literature although a discussion of the work (without proofs) appears in the Proceedings of the 1985 conference at St. Andrews [Lond. Math. Soc. Lect. Note Ser. 121, 128-134 (1986; Zbl 0607.20027)]. The author is interested in the following general problems. Let G denote a linear algebraic group. Suppose G is defined over an algebraic number field k, S is a finite set of primes of k and \(\Gamma\) is an S-arithmetic subgroup of G. How can it be determined whether \(\Gamma\) has a finite presentation? Suppose that K is a p-adic field (p a prime in S, say) so that the K-points \(G_ K\) of G are a subgroup of the general linear group over K. Since the latter is a topological group, it makes sense to refer to a compact generating set or even compact presentation for \(G_ K\). How can we determine whether \(G_ K\) has a compact presentation? These are well studied problems. \textit{A. Borel} and \textit{Harish-Chandra} [Ann. Math., II. Ser. 75, 485-535 (1962; Zbl 0107.148)] have shown that arithmetic groups are always finitely presented and \textit{H. Behr} [Invent. Math. 4, 265-274 (1967; Zbl 0157.367)] has shown that if G is reductive, then \(\Gamma\) is always finitely presented, whatever the set S of primes. In the case that K is a p-adic field, \textit{A. Borel} and \textit{J. Tits} [Publ. Math., Inst. Hautes Etud. Sci. 27, 659-755 (1965; Zbl 0145.174)] have shown that G will always have a maximal K-split solvable subgroup H so that \(G_ K/H_ K\) is compact and have given a criterion for solving the question of whether G is compactly generated. The connection between the two clauses of the title is given by a Theorem of \textit{M. Kneser} [J. Reine Angew. Math. 214/215, 345-349 (1964; Zbl 0141.026)]. Kneser shows that \(\Gamma\) is finitely generated (finitely presented) if and only if, for each prime p in S, the group \(G_{k_ p}\) of \(k_ p\)-points of G is compactly generated (compactly presented). Here \(k_ p\) denotes the completion of k with respect to the p-adic topology. This, coupled with the results of Borel and Tits, enable the question of finite presentation of S-arithmetic groups to be translated into the question of compact presentation of the group of K-points of a K-split soluble linear algebraic group, where K is a p-adic field. The new and major contribution to be found here is a criterion for deciding the latter question of compact presentation and so also the question of the finite presentation of S-arithmetic groups. The criterion is in the spirit of the criterion of Borel and Tits for compact generation and also bear a strong relation to the criterion used by \textit{R. Bieri} and \textit{R. Strebel} [Proc. Lond. Math. Soc., III. Ser. 41, 439-464 (1980; Zbl 0448.20029)] for the finite presentation of (abstract) groups. To describe the results we shall need some notation. Let K be a p-adic field and let H be a K-split solvable algebraic group. So H is the split extension of its unipotent radical U by a torus T which is K-isomorphic to a product of copies of \(GL_ 1\). Let \({\mathfrak u}\) denote the Lie algebra of U and let \({\mathfrak u}^{ab}\) denote its Abelianisation. Note that T acts on \({\mathfrak u}\) via the adjoint representation and so also acts on is Abelianisation \({\mathfrak u}^{ab}\) and its second homology with trivial coefficients \(H_ 2({\mathfrak u})\). Given any representation of T, we can, as usual, form the weights of the representation (that is, the characters of T corresponding to one-dimensional T-invariant sub-spaces of Y). We shall say that a representation is tame if no weight is zero and no two weights add up to zero. Then the main Theorem of the monograph is: The group of K-points \(H_ K\) of H has a compact presentation if and only if the following two conditions hold. a) The representation of T on \({\mathfrak u}^{ab}\) is tame. b) Zero is not a weight of the representation of T on \(H_ 2({\mathfrak u})\). (In comparison, the work of Borel and Tits implies that \(H_ K\) is compactly generated if and only if 0 is not a weight of the representation of T on \({\mathfrak u}^{ab}.)\) The proof of this theorem occupies almost half of the monograph. The proof of the ''only if'' part is relatively straightforward. In particular the necessity of the first condition is a translation (although not a trivial one) of a theorem of Bieri and Strebel for abstract groups. In the latter work, ''weights'' are replaced by a suitably chosen subset of the character group of the commutator quotient group of H. The author shows how to relate the two concepts. The proof of the reverse part of the theorem involves some new and deep ideas. The truth of condition a) is used to construct, as a colimit of certain subgroups of H, a compactly presented group \(H^*\) so that H is quotient of \(H^*\) by a central subgroup. The construction of this group \(H^*\) is neatly accomplished via the co-limit construction but there is considerable technical difficulty in showing that \(H^*\) is a solvable group with a structure similar to that of H. Since \(H^*\) is a central extension of H it is reasonable that the compact presentability of H can be interpreted in terms of the second homology. This translates, via an isomorphism theorem between group homology and Lie algebra homology, as condition b). The last two chapters are devoted to exploring the consequences of this Theorem. Because of the result of Borel and Tits, we have a criterion for deciding the compact presentability of G (albeit in terms of invariants associated with the solvable subgroup H). By Kneser's Theorem, we also have criteria for deciding the finite presentability of any S-arithmetic group. The author finishes with two major contributions to bringing the criteria a little closer to the group in question. Firstly he proves a theorem rather similar to that quoted but assuming only that H is K-split (i.e. has a maximal K-split torus) rather than that H is K-split solvable. Secondly, he gives a purely group-theoretical criterion for the finite presentability of solvable S-arithmetic groups (due originally to \textit{H. Åberg} [''Solvable groups and \(FP_ n''\), preprint]). Theorem. Let \(\Gamma\) be a solvable S-arithmetic group. Let \(\Delta\) be a nilpotent normal subgroup of \(\Gamma\) such that \(\Gamma\) /\(\Delta\) has a finitely generated abelian subgroup Q of finite index. Then \(\Gamma\) is finitely presented if and only if the following two conditions hold. a) \(\Delta^{ab}\) is a tame \({\mathbb{Z}}Q\)-module. b) \(H_ 2(\Delta,{\mathbb{Z}})\) is a finitely generated \({\mathbb{Z}}\Gamma\)-module. (Here ''tame'' is intended in the sense of Bieri and Strebel, whose work also shows that a) may be replaced by the condition that the largest metabelian quotient \(\Gamma\) /\(\Gamma\) '' is finitely presented.)
scientific article; zbMATH DE number 4006522