J. Theory 12 (2009), 555–559 Journal of Group Theory DOI 10.1515/JGT.2008.096 ( de Gruyter 2009

The compact generation of closed of locally compact groups

Karl Heinrich Hofmann and Karl-Hermann Neeb (Communicated by S. A. Morris)

Abstract. It is shown that a closed almost solvable of an almost connected locally is compactly generated. In particular, every discrete solvable subgroup of a connected locally compact group is finitely generated.

A G is said to be almost solvable ([2, p. 356]) if there is a normal solvable subgroup S such that G=S is compact. It is said to be almost connected if there is a connected normal subgroup C such that G=C is compact. We shall prove the following result.

Main Theorem. A closed almost solvable subgroup of a locally compact almost con- nected group is compactly generated.

Being compactly generated is a property of a topological group G that is preserved under the formation of arbitrary products and passage to quotients. The passage to a closed subgroup H is problematic (unless G is locally compact and G=H is compact, see [6] and [1, Chapitre 7, §3, Lemma 3]); this is where this note comes in. The essay [11] provides some background. From the Main Theorem it follows, in particular, that a discrete solvable subgroup of an almost connected locally compact group is finitely generated.

Example S. The connected 3-dimensional simple PSLð2; RÞ contains a dis- crete free group of infinite rank; such a closed subgroup is not compactly generated.

We remark that a nonabelian free group is countably nilpotent (see e.g. [5, Defini- tion 10.5]); that is, the descending central series terminates at the identity subgroup after o steps. The Main Theorem therefore fails for transfinitely solvable subgroups in place of solvable ones. A connected semisimple Lie group is noncompact if and only if the quotient modulo its center contains a copy of PSLð2; RÞ, if and only if it contains a discrete 556 K. H. Hofmann and K.-H. Neeb nonabelian free subgroup; in other words, a connected Lie group is almost solvable if and only if it does not contain a discrete nonabelian free subgroup. The following example shows that subgroups of finitely generated solvable groups need not be finitely generated:

Example SOL. Let G J Q z Q be the subgroup generated by the two elements a :¼ð0; 2Þ and b :¼ð1; 0Þ. Then 1 G G Z z Z; 2y is a 2-generator metabelian group, while the abelian subgroup

1 Z f0g 2y is not finitely generated.

Thus, in the Main Theorem, the hypothesis ‘G=G0 compact’ cannot be relaxed to ‘G=G0 compactly generated’. For abelian subgroups the Main Theorem will allow us to derive a characterization theorem for compactly generated locally compact abelian groups as follows.

Corollary 1. For a locally compact A the following conditions are equivalent: (1) A is compactly generated; (2) A G Rk l C l Zn for a unique largest compact subgroup C and natural numbers k, n; (3) the character group A^ is a Lie group; (4a) there is an almost connected locally compact abelian group G and a closed subgroup H such that A G H; (4b) there is an almost connected locally compact group G and a closed subgroup H such that A G H.

Proof. (1) ) (2) See e.g. [2, p. 346], or [4, Theorem 7.57(ii)]. c c (2) ) (3) We have A^G Rk l C^ l Zn G Rk l D l Tn for a discrete abelian group D. This is a Lie group. ^ ^ Rk l Tn (3) ) (2) If A is a Lie group, then ðAÞ0 is open and isomorphic to for some k and n; it is divisible, whence A^G ðRk l TnÞ l D for a discrete subgroup D. c c Hence A ¼ Rk l D^ l Tn G Rk l C l Zn for the unique largest compact subgroup C of A. Compactly generated subgroups of locally compact groups 557

(2) ) (4a) We have A J Rk C Rn G Rkþn l C, an almost connected locally compact abelian group. Trivially (4a) ) (4b). (4b) ) (1) Let G be an almost connected locally compact group and A a closed abelian subgroup. Then A is, in particular, solvable. Hence the Main Theorem provides the required implication. r

By comparison with Example SOL, the situation for abelian groups is distinctly simpler than it is for metabelian groups:

Corollary 2 (Morris’ theorem, 1972 [7], [11]). A closed subgroup of a compactly generated locally compact abelian group is compactly generated.

Proof. We proved (1) ) (2) , (3) in Corollary 1 independently of the Main Theo- rem. Thus if G is a locally compact compactly generated abelian group, then G^ is an abelian Lie group. The character group A^ of a closed subgroup A of G, by duality, is a quotient of the Lie group G^ and thus is a Lie group. Hence A satisfies (2) and thus is clearly compactly generated. r

A considerable generalization was obtained in 1973 in a paper by Guivarc’h ([2, III.1, p. 347, III.4, p. 352]). He calls a topological group noetherian if every closed subgroup is compactly generated.

Theorem (Theorem of Guivarc’h [2]). For a locally compact solvable group G, the following conditions are equivalent: (1) G is noetherian; (2) each factor group DnðGÞ=Dnþ1ðGÞ of the descending series of closed commutator subgroups is compactly generated. Moreover, if G is, in addition, a Lie group, then these conditions are also equivalent to

(3) G=G0 is noetherian. In particular, every closed subgroup of a locally compact connected solvable group is compactly generated.

This shows that the 2-generator metabelian group G of Example SOL cannot be realized as H=H0 for a closed subgroup H of a connected solvable Lie group G, let alone be discretely embedded into G. In [12], Wang described those solvable groups which occur as discrete subgroups of almost connected solvable Lie groups. Forerunners of the results of Guivarc’h above are to be found in the paper by Mostow (see e.g. [8, Theorem 5.5]). As we now begin a proof of the Main Theorem we first reduce it to a result on connected Lie groups and their closed solvable subgroups: 558 K. H. Hofmann and K.-H. Neeb

Reduction 1. The Main Theorem holds if every closed solvable subgroup H of a con- nected Lie group G is compactly generated.

Proof. Indeed let G be an almost connected locally compact group and N a compact normal subgroup such that G=N is a Lie group. The existence of N is a consequence of Yamabe’s theorem saying that each almost connected locally compact group is a pro-Lie group ([13], [14]). Let H be a closed almost solvable subgroup of G. Then there is a closed solvable subgroup S of H such that H=S is compact. We shall show that S is compactly generated; this will imply that H is compactly generated. Then SN is a closed subgroup and SN=N is a closed solvable subgroup A of the Lie group L ¼ G=N with finitely many components. If our claim is true for connected Lie groups G, then A V L0 is compactly generated. We may assume L ¼ L0A. Then A V L0 has finite index in A. Therefore A ¼ SN=N is compactly generated. Thus SN is compactly generated. So S is compactly generated. (See [6] or [1, Chapitre VII, §3, Lemma 3].) r

Reduction 2. The Main Theorem holds if every closed solvable subgroup H of a con- nected simple center-free Lie group G is compactly generated.

Proof. By Reduction 1 we may assume that G is a connected Lie group and H a closed solvable subgroup. Let N be that normal subgroup of G which contains the radical R of G such that N=R is the discrete center of the semisimple group G=R. Then G=N is center-free semisimple and so is a direct product S1 ...Sn of connected simple center-free groups. Let Ak be the projection into the factor Sk of the solv- able group HN=N. Let A be that closed subgroup of G containing N for which A=N ¼ A1 ...An. Assume that the Main Theorem is true for simple center-free con- nected Lie groups. Then Ak is noetherian, and so A=N is noetherian. Since N=R is finitely generated central, A=R is noetherian. Since R J A0, this implies that A is noetherian. So H J A is compactly generated. r

Lemma 3. In a connected simple center-free Lie group, any closed solvable subgroup is compactly generated.

Proof. Let G be a connected simple center-free Lie group and H a closed solvable subgroup. Via adjoint representation, G is a closed subgroup of

AutðgÞ J GLðgÞ J GLðC n gÞ:

The Zariski closure A of H in GLðC n gÞ is a complex linear algebraic group and so A=A0 is finite (see e.g. [9, Theorems 3.1.1 and 3.3.1]). Hence A satisfies hypothesis (3) of the theorem of Guivarc’h and thus is noetherian by that theorem. (See also [8, Theorem 5.5].) Hence H J A is compactly generated. r

This completes the proof of the Main Theorem. Compactly generated subgroups of locally compact groups 559

Acknowledgment. The authors posted a di¤erent proof of the Main Theorem on arXiv: math.GR.0801.4234, unaware of the article [2] by Guivarc’h. They thank Yves de Cornulier for pointing this reference; he suggested an alternative proof based on information in that paper. The present short proof is self-contained modulo [2]. In an e-mail of July 10, 2008, de Cornulier communicated references [8] and [12] which belong to the present context. A considerable body of results on solvable subgroups of Lie groups is in the litera- ture, and on properties related, but not equivalent to, being compactly generated. The precise content of the main result of this note does not appear to be among them in a readily available form.

References [1] N. Bourbaki. Inte´gration, Chapitres 7 et 8 (Hermann, 1963). [2] Y. Guivarc’h. Croissance polynomiale et pe´riodes des functions harmoniques. Bull. Soc. Math. France 101 (1973), 333–379. [3] G. Hochschild. The structure of Lie groups (Holden Day, 1965). [4] K. H. Hofmann and S. A. Morris. The structure of compact groups (de Gruyter, 1998 and 2006). [5] K. H. Hofmann and S. A. Morris. The Lie theory of connected pro-Lie groups (European Mathematical Society Publishing House, 2007). [6] A. M. MacBeath and S. S´wierzkowski. On the set of generators of a subgroup. Indag. Math. 21 (1959), 280–281. [7] S. A. Morris. Locally compact abelian groups and the variety of topological groups generated by the reals. Proc. Amer. Math. Soc. 34 (1972), 290–292. [8] G. D. Mostow. Some applications of representative functions to solvmanifolds. Amer. J. Math. 93 (1971), 11–32. [9] A. L. Onishchik and E. B. Vinberg. Lie groups and algebraic groups (Springer-Verlag, 1990). [10] M. S. Raghunathan. Discrete subgroups of Lie groups. Ergebnisse der Math. 68 (Springer- Verlag, 1972). [11] K. Ross. Closed subgroups of compactly generated LCA group are compactly generated. http://www.uoregon.edu/~ross1/subgroupsofCGLCA6.pdf, 2007. [12] H. C. Wang. Discrete subgroups of solvable Lie groups. Ann. of Math. (2) 64 (1956), 11–19. [13] H. Yamabe. On the conjecture of Iwasawa and Gleason, Ann. of Math. 58 (1953), 48–54. [14] H. Yamabe. Generalization of a theorem of Gleason. Ann. of Math. (2) 58 (1953), 351–365.

Received 31 July, 2008; revised 20 October, 2008 Karl Heinrich Hofmann, Fachbereich Mathematik, Technische Universita¨t Darmstadt, Schlossgartenstrasse 7, 64289 Darmstadt, Germany E-mail: [email protected] Karl-Hermann Neeb, Fachbereich Mathematik, Technische Universita¨t Darmstadt, Schloss- gartenstrasse 7, 64289 Darmstadt, Germany E-mail: [email protected]