Cocompact Lattice Research Articles

Residually finite non-linear hyperbolic groups

We exhibit the first examples of residually finite non-linear Gromov hyperbolic groups. Our examples are constructed as amalgamated products of torsion-free cocompact lattices in the rank 1 Lie group \mathsf{Sp}(d,1) , d \geq2 , along maximal cyclic subgroups.

Finite Groups Acting on Compact Complex Parallelizable Manifolds

Abstract We prove that the automorphism group of a compact complex parallelizable manifold is Jordan. In the course of the proof, we show that outer automorphism groups of cocompact lattices in complex Lie groups have bounded finite subgroups.

A type I conjecture and boundary representations of hyperbolic groups

AbstractWe establish new results on the weak containment of quasi‐regular and Koopman representations of a second countable locally compact group associated with nonsingular ‐spaces. We deduce that any two boundary representations of a hyperbolic locally compact group are weakly equivalent. We also show that nonamenable hyperbolic locally compact groups with a cocompact amenable subgroup are characterized by the property that any two proper length functions are homothetic up to an additive constant. Combining those results with the work of Ł. Garncarek on the irreducibility of boundary representations of discrete hyperbolic groups, we deduce that a type I hyperbolic group with a cocompact lattice contains a cocompact amenable subgroup. Specializing to groups acting on trees, we answer a question of C. Houdayer and S. Raum.

Weighted $$L^2$$ Holomorphic Functions on Ball-Fiber Bundles Over Compact Kähler Manifolds

Let $${\widetilde{M}}$$ be a complex manifold, $$\Gamma $$ be a torsion-free cocompact lattice of $$\text {Aut}({\widetilde{M}})$$ and $$\rho :\Gamma \rightarrow SU(N,1)$$ be a representation. Suppose that there exists a $$\rho $$ -equivariant totally geodesic isometric holomorphic embedding $$\imath :{{\widetilde{M}}}\rightarrow {\mathbb {B}}^N$$ . Let $$M:={{\widetilde{M}}}/\Gamma $$ and $$\Sigma :={\mathbb {B}}^N/\rho (\Gamma )$$ . In this paper, we investigate a relation between weighted $$L^2$$ holomorphic functions on the fiber bundle $$\Omega :=M\times _\rho {\mathbb {B}}^N$$ and the holomorphic sections of the pull-back bundle $$\imath ^*(S^mT^*_\Sigma )$$ over M. In particular, $$A^2_\alpha (\Omega )$$ has infinite dimension for any $$\alpha >-1$$ and if $$n<N$$ , then $$A^2_{-1}(\Omega )$$ also has the same property. As an application, if $$\Gamma $$ is a torsion-free cocompact lattice in SU(n, 1), $$n\ge 2$$ , and $$\rho :\Gamma \rightarrow SU(N,1)$$ is a maximal representation, then for any $$\alpha >-1$$ , $$A^2_\alpha ({\mathbb {B}}^n\times _{\rho } {\mathbb {B}}^N)$$ has infinite dimension. If $$n<N$$ , then $$A_{-1}^2({\mathbb {B}}^n\times _{\rho } \mathbb B^N)$$ also has the same property.

On closed manifolds admitting an Anosov diffeomorphism but no expanding map

A few years ago, the first example of a closed manifold admitting an Anosov diffeomorphism but no expanding map was given. Unfortunately, this example is not explicit and is high-dimensional, although its exact dimension is unknown due to the type of construction. In this paper, we present a family of concrete 12-dimensional nilmanifolds with an Anosov diffeomorphism but no expanding map, where a nilmanifold is defined as the quotient of a 1-connected nilpotent Lie group by a cocompact lattice. We show that this family has the smallest possible dimension in the class of infra-nilmanifolds, which is conjectured to be the only type of manifolds admitting Anosov diffeomorphisms up to homeomorphism. The proof shows how to construct positive gradings from the eigenvalues of the Anosov diffeomorphism under some additional assumptions related to the rank, using the action of the Galois group on these algebraic units.

Algebraic dimension and complex subvarieties of hypercomplex nilmanifolds

A nilmanifold is a (left) quotient of a nilpotent Lie group by a cocompact lattice. A hypercomplex structure on a manifold is a triple of complex structure operators satisfying the quaternionic relations. A hypercomplex nilmanifold is a compact quotient of a nilpotent Lie group equipped with a left-invariant hypercomplex structure. Such a manifold admits a whole 2-dimensional sphere S2 of complex structures induced by quaternions. We prove that for any hypercomplex nilmanifold M and a generic complex structure L∈S2, the complex manifold (M,L) has algebraic dimension 0. A stronger result is proven when the hypercomplex nilmanifold is abelian. Consider the Lie algebra of left-invariant vector fields of Hodge type (1,0) on the corresponding nilpotent Lie group with respect to some complex structure I∈S2. A hypercomplex nilmanifold is called abelian when this Lie algebra is abelian. We prove that all complex subvarieties of (M,L) for generic L∈S2 on a hypercomplex abelian nilmanifold are also hypercomplex nilmanifolds.

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.

Projective and conformal closed manifolds with a higher-rank lattice action

We prove global results about actions of cocompact lattices in higher-rank simple Lie groups on closed manifolds endowed with either a projective class of connections or a conformal class of pseudo-Riemannian metrics of signature (p, q), with \(\min (p,q) \ge 2\). Building on a recent article [17], provided that such a structure is locally equivalent to its model \({\textbf{X}}\), the main question treated here is the completeness of the associated \((G,{\textbf{X}})\)-structure. Because of the similarities between the model spaces of projective geometry and non-Lorentzian conformal geometry, a number of arguments apply in both contexts. We therefore present the proofs in parallel. The conclusion is that in both cases, when the real rank is maximal, the manifold is globally equivalent to either the model space \({\textbf{X}}\) or its double cover.

Zimmer's conjecture: Subexponential growth, measure rigidity, and strong property (T)

We prove several cases of Zimmer's conjecture for actions of higher-rank, cocompact lattices on low-dimensional manifolds. For example, if $\Gamma$ is a cocompact lattice in $\mathrm{SL}(n, \mathbb{R})$, $M$ is a compact manifold, and $\omega$ a volume form on $M$, we show that any homomorphism $\alpha\colon \Gamma \rightarrow \mathrm{Diff}(M)$ has finite image if the dimension of $M$ is less than $n-1$ and that any homomorphism $\alpha\colon \Gamma \rightarrow\mathrm{Diff}(M,\omega)$ has finite image if the dimension of $M$ is less than $n$. The key step in the proof is to show that any such action has uniform subexponential growth of derivatives. This is established using ideas from the smooth ergodic theory of higher-rank abelian groups, structure theory of semisimple groups, and results from homogeneous dynamics. Having established uniform subexponential growth of derivatives, we apply Lafforgue's strong property (T) to establish the existence of an invariant Riemannian metric.

Real semisimple Lie groups and balanced metrics

Given any non-compact real simple Lie group \mathrm{G}_o of inner type and even dimension, we prove the existence of an invariant complex structure \mathrm{J} and a Hermitian balanced metric on \mathrm{G}_o and on any compact quotient \mathrm{M}= \Gamma\backslash\mathrm{G}_o , with \Gamma a cocompact lattice. We also prove that (\mathrm{M},\mathrm{J}) does not carry any pluriclosed metric, in contrast to the case of even dimensional compact Lie groups, which admit pluriclosed but not balanced metrics.

Residually Finite Lattices in PU(2,1)˜ and Fundamental Groups of Smooth Projective Surfaces

This paper studies residual finiteness of lattices in the universal cover of PU(2,1) and applications to the existence of smooth projective varieties with fundamental group a cocompact lattice in PU(2,1) or a finite covering of it. First, we prove that certain lattices in the universal cover of PU(2,1) are residually finite. To our knowledge, these are the first such examples. We then use residually finite central extensions of torsion-free lattices in PU(2,1) to construct smooth projective surfaces that are not birationally equivalent to a smooth compact ball quotient but whose fundamental group is a torsion-free cocompact lattice in PU(2,1).

Homological methods in certain Picard group computations

Let G be a connected complex semisimple Lie group, Γ be a cocompact, irreducible and torsionless lattice in G and K be a maximal compact subgroup of G. Assume Γ acts by left multiplication and K acts by right multiplication on G. Let MΓ=Γ﹨G, X=G/K and XΓ=Γ﹨X. In this article we prove that for any n≥0, the composition Hn(XΓ,C)→Hn(MΓ,C)→Hn(MΓ,OMΓ) is an isomorphism. As an application when G is simply connected, we compute the Picard group of MΓ for the cases rank(G) =1,2. More precisely we show that if rank(G) =1, Pic(MΓ)=(Cr/Zr)⊕A and if rank(G) =2, then Pic(MΓ)≅A via the first Chern class map, where A is the torsion subgroup of H2(MΓ,Z) and r is the rank of Γ/[Γ,Γ].

Lattices in a product of trees, hierarchically hyperbolic groups and virtual torsion-freeness.

We construct cocompact lattices in a product of trees which are not virtually torsion‐free. This gives the first examples of hierarchically hyperbolic groups which are not virtually torsion‐free.

Asymptotic total geodesy of local holomorphic curves exiting a bounded symmetric domain and applications to a uniformization problem for algebraic subsets

The current article stems from our study on the asymptotic behavior of holomorphic isometric embeddings of the Poincaré disk into bounded symmetric domains. As a first result we prove that any holomorphic curve exiting the boundary of a bounded symmetric domain $\Omega$ must necessarily be asymptotically totally geodesic. Assuming otherwise we derive by the method of rescaling a hypothetical holomorphic isometric embedding of the Poincaré disk with $\operatorname{Aut}(\Omega^\prime)$-equivalent tangent spaces into a tube domain $\Omega^\prime \subset \Omega$ and derive a contradiction by means of the Poincaré–Lelong equation. We deduce that equivariant holomorphic embeddings between bounded symmetric domains must be totally geodesic. Furthermore, we solve a uniformization problem on algebraic subsets $Z \subset \Omega$. More precisely, if $\check{\Gamma} \subset \operatorname{Aut} (\Omega)$ is a torsion-free discrete subgroup leaving $Z$ invariant such that $Z / \check{\Gamma}$ is compact, we prove that $Z \subset \Omega$ is totally geodesic. In particular, letting $\Gamma \subset \operatorname{Aut} (\Omega)$ be a torsion-free cocompact lattice, and $\pi : \Omega \to \Omega / \Gamma =: X_\Gamma$ be the uniformization map, a subvariety $Y \subset X_\Gamma$ must be totally geodesic whenever some (and hence any) irreducible component $Z$ of $\pi-(Y)$ is an algebraic subset of $\Omega$. For cocompact lattices this yields a characterization of totally geodesic subsets of $X_\Gamma$ by means of bi-algebraicity without recourse to the celebrated monodromy result of André–Deligne on subvarieties of Shimura varieties, and as such our proof applies to not necessarily arithmetic cocompact lattices. In place of the monodromy result of André–Deligne we exploit the existence theorem of Aubin and Yau on Kähler–Einstein metrics for projective manifolds $Y$ satisfying $c_1 (Y) \lt 0$ and make use of Nadel’s semisimplicity theorem on automorphism groups of noncompact Galois covers of such manifolds, together with the total geodesy of equivariant holomorphic isometric embeddings between bounded symmetric domains.

Arithmetic groups and the Lehmer conjecture

We generalize a result of Sury [Sur92] and prove that uniform discreteness of cocompact lattices in higher rank semisimple Lie groups (first conjectured by Margulis [Mar91]) is equivalent to a weak form of Lehmer’s conjecture. We include a short survey of related results and conjectures.

Positive Hermitian curvature flow on nilpotent and almost-abelian complex Lie groups

We study the positive Hermitian curvature flow on the space of left-invariant metrics on complex Lie groups. We show that in the nilpotent case, the flow exists for all positive times and subconverges in the Cheeger-Gromov sense to a soliton. We also show convergence to a soliton when the complex Lie group is almost abelian. That is, when its Lie algebra admits a (complex) co-dimension one abelian ideal. Finally, we study solitons in the almost-abelian setting. We prove uniqueness and completely classify all left-invariant, almost-abelian solitons, giving a method to construct examples in arbitrary dimensions, many of which admit co-compact lattices.

On the vanishing of reduced 1-cohomology for Banach representations

A theorem of Delorme states that every unitary representation of a connected Lie group with nontrivial reduced first cohomology has a finite-dimensional subrepresentation. More recently Shalom showed that such a property is inherited by cocompact lattices and stable under coarse equivalence among amenable countable discrete groups. We give a new geometric proof of Delorme's theorem which extends to a larger class of groups, including solvable $p$-adic algebraic groups, and finitely generated solvable groups with finite Pr\"ufer rank. Moreover all our results apply to isometric representations in a large class of Banach spaces, including reflexive Banach spaces. As applications, we obtain an ergodic theorem in for integrable cocycles, as well as a new proof of Bourgain's Theorem that the 3-regular tree does not embed quasi-isometrically into any superreflexive Banach space.

Subconvex bounds for Hecke\u2013Maass forms on compact arithmetic quotients of semisimple Lie groups

Let H be a semisimple algebraic group, K a maximal compact subgroup of G:=H({{mathbb {R}}}), and Gamma subset H({{mathbb {Q}}}) a congruence arithmetic subgroup. In this paper, we generalize existing subconvex bounds for Hecke–Maass forms on the locally symmetric space Gamma backslash G/K to corresponding bounds on the arithmetic quotient Gamma backslash G for cocompact lattices using the spectral function of an elliptic operator. The bounds obtained extend known subconvex bounds for automorphic forms to non-trivial K-types, yielding such bounds for new classes of automorphic representations. They constitute subconvex bounds for eigenfunctions on compact manifolds with both positive and negative sectional curvature. We also obtain new subconvex bounds for holomorphic modular forms in the weight aspect.

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.

Cocompact lattices in locally pro--complete rank-2 Kac-Moody groups

We initiate an investigation of lattices in a new class of locally compact groups: so-called locally pro--complete Kac-Moody groups. We discover that in rank 2 their cocompact lattices are particularly well- behaved: under mild assumptions, a cocompact lattice in this completion contains no elements of order . This statement is still an open question for the Caprace-Rémy-Ronan completion. Using this, modulo results of Capdeboscq and Thomas, we classify edge-transitive cocompact lattices and describe a cocompact lattice of minimal covolume. Bibliography: 22 titles.