Lattices In SL Research Articles

Constructing thin subgroups of SL(n + 1, ℝ) via bending

In this paper we use techniques from convex projective geometry to produce many new examples of thin subgroups of lattices in special linear groups that are isomorphic to the fundamental groups of finite volume hyperbolic manifolds. More specifically, we show that for a large class of arithmetic lattices in SO(n,1) it is possible to find infinitely many non-commensurable lattices in SL(n+1,R) that contain a thin subgroup isomorphic to a finite index subgroup of the original arithmetic lattice. This class of arithmetic lattices includes all non-cocompact arithmetic lattices and all cocompact arithmetic lattices when $n$ is even.

ON THE TORSION HOMOLOGY OF NON-ARITHMETIC HYPERBOLIC TETRAHEDRAL GROUPS

Numerical data concerning the growth of torsion in the first homology of congruence subgroups of non-arithmetic hyperbolic tetrahedral groups are collected. The data provide support the speculations of Bergeron and Venkatesh on the growth of torsion homology and the regulators for lattices in SL 2(ℂ).

Noncommutative Lp-spaces without the completely bounded approximation property

For any 1\leq p \leq \infty different from 2, we give examples of non-commutative Lp spaces without the completely bounded approximation property. Let F be a non-archimedian local field. If p>4 or p<4/3 and r\geq 3 these examples are the non-commutative Lp-spaces of the von Neumann algebra of lattices in SL_r(F) or in SL_r(\R). For other values of p the examples are the non-commutative Lp-spaces of the von Neumann algebra of lattices in SL_r(F) for r large enough depending on p. We also prove that if r \geq 3 lattices in SL_r(F) or SL_r(\R) do not have the Approximation Property of Haagerup and Kraus. This provides examples of exact C^*-algebras without the operator space approximation property.

Distribution results for lattices in SL(2,ℚp)

In this paper we study the ergodic properties of the linear action of lattices Γ of SL(2,ℚ p ) on ℚ p × ℚ p and distribution results for orbits of Γ. Following Serre, one can define a “geodesic flow” for an associated tree (actually associated to GL(2,ℚ p )). The approach we use is based on an extension of this approach to “frame flows” which are a natural compact group extension of the geodesic flow.

On the non-vanishing of the first Betti number of hyperbolic three manifolds

We show the non-vanishing of cohomology groups of sufficiently small congruence lattices in SL(1,D), where D is a quaternion division algebra defined over a number field E contained inside a solvable extension of a totally real number field. As a corollary, we obtain new examples of compact, arithmetic, hyperbolic three manifolds, with non-torsion first homology group, confirming a conjecture of Waldhausen. The proof uses the characterisation of the image of solvable base change by the author, and the construction of cusp forms with non-zero cusp cohomology by Labesse and Schwermer.

Deformations of complex structures on Γ/SL 2(C)

LetG be a connected complex semisimple Lie group. Let Γ be a cocompact lattice inG. In this paper, we show that whenG isSL 2(C), nontrivial deformations of the canonical complex structure onX exist if and only if the first Betti number of the lattice Γ is non-zero. It may be remarked that for a wide class of arithmetic groups Γ, one can find a subgroup Γ′ of finite index in Γ, such that Γ′/[Γ′,Γ′] is finite (it is a conjecture of Thurston that this is true for all cocompact lattices inSL(2, C)).

Connection preserving actions of lattices inSL nℝ

We consider connection-preserving actions of lattice subgroups ofSL nℝ on compact Riemannian manifolds, forn≥3. The main result gives a description of such actions, when dimM=n+1, as follows: Either the action preserves a smooth invariant Riemannian metric onM or it can be described in terms of linear actions on then-torus.

Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces

We show that ifG is a semisimple algebraic group defined overQ and Γ is an arithmetic lattice inG:=GR with respect to theQ-structure, then there exists a compact subsetC ofG/Γ such that, for any unipotent one-parameter subgroup {ut} ofG and anyg∈G, the time spent inC by the {ut}-trajectory ofgΓ, during the time interval [0,T], is asymptotic toT, unless {g−1utg} is contained in aQ-parabolic subgroup ofG. Some quantitative versions of this are also proved. The results strengthen similar assertions forSL(n,Z),n≥2, proved earlier in [5] and also enable verification of a technical condition introduced in [7] for lattices inSL(3,R), which was used in our proof of Raghunathan’s conjecture for a class of unipotent flows, in [8].

Uniformly distributed orbits of certain flows on homogeneous spaces

is non-compact. For a co-compact lattice, this result was strengthened by F/irstenberg [1 lJ proving that every orbit is uniformly distributed with respect to a G-invariant measure. For non-uniform lattices in SL(2, R), using a classification of invariant measures obtained by Dani in [21, Dani and Smillie [31 proved that every non-periodic orbit is uniformly distributed. There are also various results obtained on orbit closures and invariant measures etc. of larger subgroups consisting of unipotent elements, especially the horospherical subgroups. Recently, there was a spurt in the area initiated by Margulis' proof (cf. [151, see also [7]) of Oppenheim conjecture on values of quadratic forms at integral points using the study of unipotent flows. The reader is referred to the survey articles by Dani [41 and Margulis [14J for various related developments. We now note some conjectures expected to hold for orbits of a unipotent flow, namely the U-action on

Group Presentation, p-Adic Analytic Groups and Lattices in SL 2 ( C )

groups by using their pro-p completions. This method is based on the observation that if (X; R) is a presentation of the finitely generated (f.g.) group F, then (X; R), considered as a presentation for a pro-p group is actually a presentation for T,;, the pro-p completion of F. First we prove a generalization of a theorem of Golod and Shafarevich (cf. [H]). THEOREM. Let G be a p-adic analytic pro-p group different from Zp (= the p-adic integers). If (X; R) is a minimal presentation of G then 1X;2 IRI? 2