Non-cocompact Lattice Research Articles

On absolutely profinitely solitary lattices in higher rank Lie groups

We establish conditions under which lattices in certain simple Lie groups are profinitely solitary in the absolute sense, so that the commensurability class of the profinite completion determines the commensurability class of the group among finitely generated residually finite groups. While cocompact lattices are typically not absolutely solitary, we show that noncocompact lattices in Sp ⁡ ( n , R ) \operatorname {Sp}(n,\mathbb {R}) , G 2 ( 2 ) G_{2(2)} , E 8 ( C ) E_8(\mathbb {C}) , F 4 ( C ) F_4(\mathbb {C}) , and G 2 ( C ) G_2(\mathbb {C}) are absolutely solitary if a well-known conjecture on Grothendieck rigidity is true.

Presentations for the Euclidean Picard modular groups

Mark and Paupert devised a general method for obtaining presentations for arithmetic non-cocompact lattices, $$\Gamma $$ , in isometry groups of negatively curved symmetric spaces. The method involves a classical theorem of Macbeath applied to a $$\Gamma $$ -invariant covering by horoballs of the negatively curved symmetric space upon which $$\Gamma $$ acts. In this paper, we will discuss the application of their method to the Picard modular groups, PU $$(2,1;{\mathcal {O}}_{d})$$ , when $$d=2,11$$ , and obtain presentations for these groups, which completes the list of presentations for Picard modular groups whose entries lie in Euclidean domains, namely those with $$d=1,2,3,7,11$$ .

Strong property (T) for higher rank lattices

We prove that every lattice in a product of higher rank simple Lie groups or higher rank simple algebraic groups over local fields has Vincent Lafforgue's strong property (T). Over non-archimedean local fields, we also prove that they have strong Banach proerty (T) with respect to all Banach spaces with nontrivial type, whereas in general we obtain such a result with additional hypotheses on the Banach spaces. The novelty is that we deal with non-cocompact lattices, such as $\mathrm{SL}_n(\mathbf{Z})$ for $n \geq 3$. To do so, we introduce a stronger form of strong property (T) which allows us to deal with more general objects than group representations on Banach spaces that we call two-step representations, namely families indexed by a group of operators between different Banach spaces that we can compose only once. We prove that higher rank groups have this property and that this property passes to undistorted lattices.

About the obstacle to proving the Baum–Connes conjecture without coefficient for a non-cocompact lattice in $Sp_4$ in a local field

We introduce property (T_\mathrm {Schur},G,K) and prove it for some non-cocompact lattice in Sp_4 in a local field of finite characteristic. We show that property (T_\mathrm {Schur},G,K) for a non-cocompact lattice \Gamma in a higher rank almost simple algebraic group in a local field is an obstacle to proving the Baum Connes conjecture without coefficient for \Gamma with known methods, and this is stronger than the well-known fact that \Gamma does not have the property of rapid decay (property (RD)). It is the first example (as announced in [7]) for which all known methods for proving the Baum–Connes conjecture without coefficient fail.

Generalized Dedekind sums and equidistribution mod 1

Dedekind sums are well-studied arithmetic sums, with values uniformly distributed mod 1. Based on their relation to certain modular forms, Dedekind sums may be defined as functions on the cusp set of SL(2,Z). We present a compatible notion of Dedekind sums, which we name Dedekind symbols, for any non-cocompact lattice Γ<SL(2,R), and prove the corresponding equidistribution mod 1 result. The latter part builds up on a paper of Vardi, who first connected exponential sums of Dedekind sums to Kloosterman sums.

Lattices in complete rank 2 Kac–Moody groups

Let Λ be a minimal Kac–Moody group of rank 2 defined over the finite field Fq, where q=pa with p prime. Let G be the topological Kac–Moody group obtained by completing Λ. An example is G=SL2(K), where K is the field of formal Laurent series over Fq. The group G acts on its Bruhat–Tits building X, a tree, with quotient a single edge. We construct new examples of cocompact lattices in G, many of them edge-transitive. We then show that if cocompact lattices in G do not contain p-elements, the lattices we construct are the only edge-transitive lattices in G, and that our constructions include the cocompact lattice of minimal covolume in G. We also observe that, with an additional assumption on p-elements in G, the arguments of Lubotzky (1990) [21] for the case G=SL2(K) may be generalised to show that there is a positive lower bound on the covolumes of all lattices in G, and that this minimum is realised by a non-cocompact lattice, a maximal parabolic subgroup of Λ.

Infinite generation of non-cocompact lattices on right-angled buildings

Let \Gamma be a non-cocompact lattice on a locally finite regular right-angled building X. We prove that if \Gamma has a strict fundamental domain then \Gamma is not finitely generated. We use the separation properties of subcomplexes of X called tree-walls.

Some geometric groups with Rapid Decay

We explain simple methods to establish the property of Rapid Decay for a number of groups arising geometrically. Those lead to new examples of groups with the property of Rapid Decay, notably including non-cocompact lattices in rank one Lie groups.

New properties of lattices in Lie groups

We study finite extension groups of lattices in Lie groups which have finitely many connected components. We show that every non-cocompact Fuchsian group (these are the non-cocompact lattices in PSL (2, R) ) has an extension group of finite index which is not isomorphic to a lattice in a Lie group with finitely many connected components. On the other hand we prove that these are, in an appropriate sense, the only lattices in Lie groups which have extension groups of this kind. We also show that an extension group of finite index of a lattice in a Lie group with finitely many connected components has only finitely many conjugacy classes of finite subgroups. To cite this article: F. Grunewald, V. Platonov, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

A remark about actions of lattices on free groups

Every homomorphism from an irreducible, noncocompact lattice in a higher-rank semisimple Lie group to the outer automorphism group of a free group must have a finite image.

Finitely generated profinitely dense free groups in higher rank semi-simple groups

We prove that if Γ is an arithmetic subgroup of a non-compact linear semi-simple groupG such that the associated simply connected algebraic group over ℚ has the so-called congruence subgroup property, then Γ contains a finitely generated profinitely dense free subgroup. As a corollary we obtain af·g·p·d·f subgroup of SL n (ℤ) (n ≧ 3. More generally, we prove that if Γ is an irreducible arithmetic non-cocompact lattice in a higher rank group, then Γ containsf·g·p·d·f groups.

Quasi-flats and rigidity in higher rank symmetric spaces

In this paper we use elementary geometrical and topological methods to study some questions about the coarse geometry of symmetric spaces. Our results are powerful enough to apply to noncocompact lattices in higher rank symmetric spaces, such as SL(n, 2), n > 3: Theorem 8.1 is a major step towards the proof of quasiisometric rigidity of such lattices ([E]). We also give a different, and effective, proof of the theorem of Kleiner-Leeb on the quasi-isometric rigidity of higher rank symmetric spaces ([KL]).

On systems of generators of arithmetic subgroups of higher rank groups

We show that any two maximal disjoint unipotent subgroups of an irreducible non-cocompact lattice in a Lie group of rank atleast two generates a lattice. The proof uses techniques of the solution of the congruence subgroup problem. We show that any two maximal opposing unipotent subgroups of an irreducible lattice in a higher rank Lie Group, generate a lattice in the Lie Group. The method of proof is to use certain techniques of the solution of the congruence subgroup problem of arithmetic lattices in higher rank groups. We freely use the notation and results of [3] without giving explicit references therein. Let G be a simply connected absolutely almost simple linear algebraic group defined and isotropic over a global field K. Let C/+ be the unimpotent radical (which is defined over K) of a minimal parabolic A'-subgroup P+ of G. Let U~~ be the unipotent radical of another minimal parabolic A'-subgroup P~ of G which is opposed to P+ in the sense that U+ Γ\U~ = {1}. Let S be a finite set of places of including all the archimedian ones, if any. We call thering A — Os — {x G K \x v < 1 for all places v of A', not in S} the ring of S-integers in K. Choose a faithful representation G <—» GLpj defined over and define G(Os) = {g G G gij G Os, 1 < i,j < N}. The subgroups in G which are of finite index in G(Os) are called S-arithmetic groups. Define the 5-rank of G to be the sum Kv — rank(G). Given a non-zero ideal α of A and an algebraic ves