Irreducible Lattice Research Articles

Character rigidity for lattices and commensurators

abstract: We prove an operator algebraic superrigidity statement for homomorphisms of irreducible lattices, and also their commensurators, from certain higher-rank groups into unitary groups of finite factors. This extends the authors' previous work regarding non-free measure-preserving actions, and also answers a question of Connes for such groups.

Stabilizers of stationary actions of lattices in semisimple groups

Every stationary action of a strongly irreducible lattice or commensurator of such a lattice in a general semisimple group, with at least one higher-rank connected factor, either has finite stabilizers almost surely or finite index stabilizers almost surely. Consequently, every minimal action of such a lattice on an infinite compact metric space is topologically free.

Radial rapid decay does not imply rapid decay

We provide a new, dynamical criterion for the radial rapid decay property. We work out in detail the special case of the group $\Gamma := \mathbf{SL}_2(A)$, where $A := \mathbb{F}_q[X,X^{-1}]$ is the ring of Laurent polynomials with coefficients in $\mathbb{F}_q$, endowed with the length function coming from a natural action of $\Gamma$ on a product of two trees, to show that is has the radial rapid decay (RRD) property and doesn't have the rapid decay (RD) property. The criterion also applies to irreducible lattices in semisimple Lie groups with finite center endowed with a length function defined with the help of a Finsler metric. These examples answer a question asked by Chatterji and moreover show that, unlike the RD property, the RRD property isn't inherited by open subgroups.

The noncommutative factor theorem for lattices in product groups

We prove a noncommutative Bader-Shalom factor theorem for lattices with dense projections in product groups. As an application of this result and our previous works, we obtain a noncommutative Margulis factor theorem for all irreducible lattices Γ<G in higher rank semisimple algebraic groups. Namely, we give a complete description of all intermediate von Neumann subalgebras L(Γ)⊂M⊂L(Γ↷G/P) sitting between the group von Neumann algebra and the group measure space von Neumann algebra associated with the action on the Furstenberg-Poisson boundary.

Cocycle superrigidity for profinite actions of irreducible lattices

Let \Gamma be an irreducible lattice in a product of two locally compact groups and assume that \Gamma is densely embedded in a profinite group K . We give necessary conditions which imply that the left translation action \Gamma \curvearrowright K is “virtually” cocycle superrigid: any cocycle {w\colon \Gamma\times K\rightarrow\Delta} with values in a countable group \Delta is cohomologous to a cocycle which factors through the map \Gamma\times K\rightarrow\Gamma\times K_0 for some finite quotient group K_0 of K . As a corollary, we deduce that any ergodic profinite action of \Gamma=\mathrm{SL}_2(\mathbb Z[S^{-1}]) is virtually cocycle superrigid and virtually W ^* -superrigid for any finite nonempty set of primes S .

On invariant subalgebras of group and von Neumann algebras

AbstractGiven an irreducible lattice$\Gamma $in the product of higher rank simple Lie groups, we prove a co-finiteness result for the$\Gamma $-invariant von Neumann subalgebras of the group von Neumann algebra$\mathcal {L}(\Gamma )$, and for the$\Gamma $-invariant unital$C^*$-subalgebras of the reduced group$C^*$-algebra$C^*_{\mathrm {red}}(\Gamma )$. We use these results to show that: (i) every$\Gamma $-invariant von Neumann subalgebra of$\mathcal {L}(\Gamma )$is generated by a normal subgroup; and (ii) given a weakly mixing unitary representation$\pi $of$\Gamma $, every$\Gamma $-equivariant conditional expectation on$C^*_\pi (\Gamma )$is the canonical conditional expectation onto the$C^*$-subalgebra generated by a normal subgroup.

Picard groups of certain compact complex parallelizable manifolds and related spaces

Let G be a complex simply connected semisimple Lie group and let Γ be a torsionless uniform irreducible lattice in G. Then Γ﹨G is a compact complex non-Kähler manifold whose tangent bundle is holomorphically trivial. In this note we compute the Picard group of Γ﹨G when rank(G)≥3. When rank(G)<3, we determine the group Pic0(Γ﹨G)⊂Pic(Γ﹨G) of topologically trivial holomorphic line bundles. When rank(G)≥2, we also show that Pic0(PΓ) is isomorphic to Pic0(Y) where PΓ is a Γ﹨G-bundle associated to a principal G-bundle over a compact connected complex manifold Y, and, when rank(G)≥3, we show that Pic(Y)→Pic(PΓ) is injective with finite cokernel.

Primitive lattice varieties

A variety is primitive if every subquasivariety is equational, i.e. a subvariety. In this paper, we explore the connection between primitive lattice varieties and Whitman’s condition (W). For example, if every finite subdirectly irreducible lattice in a locally finite variety [Formula: see text] satisfies Whitman’s condition (W), then [Formula: see text] is primitive. This allows us to construct infinitely many sequences of primitive lattice varieties, and to show that there are [Formula: see text] such varieties. Some lattices that fail (W) also generate primitive varieties. But if I is a (W)-failure interval in a finite subdirectly irreducible lattice [Formula: see text], and [Formula: see text] denotes the lattice with I doubled, then [Formula: see text] is never primitive.

Stationary characters on lattices of semisimple Lie groups

We show that stationary characters on irreducible lattices Gamma < G of higher-rank connected semisimple Lie groups are conjugation invariant, that is, they are genuine characters. This result has several applications in representation theory, operator algebras, ergodic theory and topological dynamics. In particular, we show that for any such irreducible lattice Gamma < G, the left regular representation lambda _{Gamma } is weakly contained in any weakly mixing representation pi . We prove that for any such irreducible lattice Gamma < G, any Uniformly Recurrent Subgroup (URS) of Gamma is finite, answering a question of Glasner–Weiss. We also obtain a new proof of Peterson’s character rigidity result for irreducible lattices Gamma < G. The main novelty of our paper is a structure theorem for stationary actions of lattices on von Neumann algebras.

Prime, irreducible, and completely irreducible lattice preradicals on modular complete lattices (I)

Based on the concept of a lattice preradical recently introduced in [T. Albu and M. Iosif, Lattice preradicals with applications to Grothendieck categories and torsion theories, J. Algebra 444 (2015) 339–366], we present and investigate in this paper, the latticial counterparts of the concepts of prime and irreducible preradicals on the category Mod-[Formula: see text] of all unital right [Formula: see text]-modules over an associative ring [Formula: see text] with identity, introduced and studied in [F. Raggi, J. Ríos Montes, H. Rincón, R. Fernández-Alonso and C. Signoret, Prime and irreducible preradicals, J. Algebra Appl. 4 (2005) 451–466].

Kakutani equivalence of unipotent flows

We study Kakutani equivalence in the class of unipotent flows acting on finite-volume quotients of semisimple Lie groups. For every such flow, we compute the Kakutani invariant of M. Ratner, the value being explicitly given by the Jordan block structure of the unipotent element generating the flow. This, in particular, answers a question of M. Ratner. Moreover, it follows that the only loosely Kronecker unipotent flows are given by (1t 01)×id acting on (SL(2,R)×G′)∕Γ′, where Γ′ is an irreducible lattice in SL(2,R)×G′ (with the possibility that G′={e}).

Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups, II

AbstractLet $M\stackrel {\rho _0}{\curvearrowleft }S$ be a $C^\infty $ locally free action of a connected simply connected solvable Lie group S on a closed manifold M. Roughly speaking, $\rho _0$ is parameter rigid if any $C^\infty $ locally free action of S on M having the same orbits as $\rho _0$ is $C^\infty $ conjugate to $\rho _0$ . In this paper we prove two types of result on parameter rigidity.First let G be a connected semisimple Lie group with finite center of real rank at least $2$ without compact factors nor simple factors locally isomorphic to $\mathop {\mathrm {SO}}\nolimits _0(n,1)(n\,{\geq}\, 2)$ or $\mathop {\mathrm {SU}}\nolimits (n,1)(n\geq 2)$ , and let $\Gamma $ be an irreducible cocompact lattice in G. Let $G=KAN$ be an Iwasawa decomposition. We prove that the action $\Gamma \backslash G\curvearrowleft AN$ by right multiplication is parameter rigid. One of the three main ingredients of the proof is the rigidity theorems of Pansu, and Kleiner and Leeb on the quasi-isometries of Riemannian symmetric spaces of non-compact type.Secondly we show that if $M\stackrel {\rho _0}{\curvearrowleft }S$ is parameter rigid, then the zeroth and first cohomology of the orbit foliation of $\rho _0$ with certain coefficients must vanish. This is a partial converse to the results in the author’s [Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups. Geom. Topol. 21(1) (2017), 157–191], where we saw sufficient conditions for parameter rigidity in terms of vanishing of the first cohomology with various coefficients.

Quasi-isometric embeddings of symmetric spaces and lattices: reducible case

We study quasi-isometric embeddings of symmetric spaces and non-uniform irreducible lattices in semi-simple higher rank Lie groups. We show that any quasi-isometric embedding between symmetric spaces of the same rank can be decomposed into a product of quasi-isometric embeddings into irreducible symmetric spaces. We thus extend earlier rigidity results about quasi-isometric embeddings to the setting of semi-simple Lie groups. We also present some examples when the rigidity does not hold, including first examples in which every flat is mapped into multiple flats.

Arithmeticity of discrete subgroups containing horospherical lattices

Let $G$ be a semisimple real algebraic Lie group of real rank at least two and $U$ be the unipotent radical of a non-trivial parabolic subgroup. We prove that a discrete Zariski dense subgroup of $G$ that contains an irreducible lattice of $U$ is an arithmetic lattice of $G$. This solves a conjecture of Margulis and extends previous work of Hee Oh.

Simple groups and irreducible lattices in wreath products

We consider the finitely generated groups acting on a regular tree with almost prescribed local action. We show that these groups embed as cocompact irreducible lattices in some locally compact wreath products. This provides examples of finitely generated simple groups quasi-isometric to a wreath product $C\wr F$, where $C$ is a finite group and $F$ a non-abelian free group.

Super-rigidity and non-linearity for lattices in products

We prove a super-rigidity result for algebraic representations over complete fields of irreducible lattices in products of groups and lattices with dense commensurator groups. We derive criteria for the non-linearity of such groups.

Superrigidity of actions on finite rank median spaces

Finite rank median spaces are a simultaneous generalisation of finite dimensional CAT(0) cube complexes and real trees. If Γ is an irreducible lattice in a product of rank-one simple Lie groups, we show that every action of Γ on a complete, finite rank median space has a global fixed point. This is in sharp contrast with the behaviour of actions on infinite rank median spaces, where even proper cocompact actions can arise.The fixed point property is obtained as corollary to a superrigidity result; the latter holds for irreducible lattices in arbitrary products of compactly generated topological groups.We exploit Roller compactifications of median spaces; these were introduced in [48] and generalise a well-known construction in the case of cube complexes. We show that the Haagerup cocycle provides a reduced 1-cohomology class that detects group actions with a finite orbit in the Roller compactification. This is new even for CAT(0) cube complexes and has interesting consequences involving Shalom's property HFD. For instance, in Gromov's density model, random groups at low density do not have HFD.

Twisted conjugacy and quasi-isometric rigidity of irreducible lattices in semisimple lie groups

Let G be a non-compact semisimple Lie group with finite centre and finitely many connected components. We show that any finitely generated group Γ which is quasi-isometric to an irreducible lattice in G has the R∞-property, namely, that there are infinitely many ϕ-twisted conjugacy classes for every automorphism ϕ of Γ. Also, we show that any lattice in G has the R∞-property, extending our earlier result for irreducible lattices.

Shrinking target problems for flows on homogeneous spaces

We study shrinking targets problems for discrete time flows on a homogenous space $\Gamma\backslash G$ with $G$ a semisimple group and $\Gamma$ an irreducible lattice. Our results apply to both diagonalizable and unipotent flows, and apply to very general families of shrinking targets. As a special case, we establish logarithm laws for cusp excursions of unipotent flows answering a question of Athreya and Margulis.

Indicability, residual finiteness, and simple subquotients of groups acting on trees

We establish three independent results on groups acting on trees. The first implies that a compactly generated locally compact group which acts continuously on a locally finite tree with nilpotent local action and no global fixed point is virtually indicable; that is to say, it has a finite index subgroup which surjects onto $\mathbf{Z}$. The second ensures that irreducible cocompact lattices in a product of non-discrete locally compact groups such that one of the factors acts vertex-transitively on a tree with a nilpotent local action cannot be residually finite. This is derived from a general result, of independent interest, on irreducible lattices in product groups. The third implies that every non-discrete Burger-Mozes universal group of automorphisms of a tree with an arbitrary prescribed local action admits a compactly generated closed subgroup with a non-discrete simple quotient. As applications, we answer a question of D. Wise by proving the non-residual finiteness of a certain lattice in a product of two regular trees, and we obtain a negative answer to a question of C. Reid, concerning the structure theory of locally compact groups.