Let $G$ be a connected Lie group. Let $\Gamma$ a lattice in $G$ not necessarily uniform (cocompact). Is it true that $\Gamma$ is finitely generated?
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2$\begingroup$ I think one old ingredient is that the lattice meets the connected amenable radical in a cocompact lattice. This being said, this reduces to the semisimple case (without compact factor). In this case, it meets the center in a finite index subgroup, which reduces to lattices in semisimple groups with no compact factor and trivial center. Then one reduces to irreducible lattices. Kazhdan's property T allows to conclude when at least one factor has Property T. In the remaining case we have a product of rank 1 groups. Maybe the case of 1 vs several factors is done separately. $\endgroup$– YCorCommented Jul 29, 2023 at 20:25
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1$\begingroup$ @YCor That's why I find Gelander & Slutsky's uniform proof interesting. $\endgroup$– Dave BensonCommented Jul 29, 2023 at 21:33
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2$\begingroup$ @DaveBenson But this paper quickly relies on essential steps of my above sketch. Namely Auslander's theorem that the lattice meets the (nonconnected) amenable radical in a lattice. This directly reduces to the case of semisimple groups with trivial center. One this is said, possibly there's a unified proof; now everything should be in Gelander's previous 2011 paper arxiv.org/abs/1102.3574. (But I would have guess that for semisimple groups Borel-Serre already provides a uniform proof.) Anyway I insist that these are separate (important) steps in the claimed "unified" proof. $\endgroup$– YCorCommented Jul 30, 2023 at 3:32
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2$\begingroup$ By the way these approaches (inspired by Borel-Serre: retracting the symmetric space by deformation) usually prove better than finite generation, namely that such lattices have a finite $K(G,1)$ — and actually one that is a compact manifold (with boundary). $\endgroup$– YCorCommented Jul 30, 2023 at 3:44
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1$\begingroup$ I had in mind (among others?) A. Borel and J.-P. Serre, Corners and arithmetic groups, Comment. Math. Helv., 48 (1973), pp. 436–491. With an appendix: Arrondissement des variétés à coins, par A. Douady et L. Hérault. The latter addresses arithmetic groups. $\endgroup$– YCorCommented Jul 30, 2023 at 11:36
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J. Lie Theory 30 (2020), no. 1, 33–40, arXiv:1903.04828 Gelander and Slutsky, "On the minimal size of a generating set for lattices in Lie groups", Corollary 1.6 says yes, and says it is "well known but non-trivial".
answered Jul 29, 2023 at 18:44
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$\begingroup$ Thank you @Dave_Benson! I am wondering where the theorem was proven the first time? $\endgroup$– M. HanCommented Jul 29, 2023 at 18:51
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1$\begingroup$ Originally a combination of bits and pieces. $\endgroup$ Commented Jul 29, 2023 at 18:59