Higher Rank Semisimple Lie Groups Research Articles

Arbeitsgemeinschaft: Higher Rank Teichmüller Theory

Higher rank Teichmüuller theory is the study of certain connected components of character varieties of surface groups in higher rank semisimple Lie groups, with the property that all elements in these components correspond to faithful representations with discrete image. Like classical Teichmüller theory, this relatively recent theory is very rich and builds on a combination of methods from various areas of mathematics. Its many facets were explored in detail during the Arbeitsgemeinschaft.

Invariant measures and measurable projective factors for actions of higher-rank lattices on manifolds

We consider smooth actions of lattices in higher-rank semisimple Lie groups on manifolds. We define two numbers $r(G)$ and $m(G)$ associated with the roots system of the Lie algebra of a Lie group $G$. If the dimension of the manifold is smaller than $r(G)$, then we show the action preserves a Borel probability measure. If the dimension of the manifold is at most $m(G)$, we show there is a quasi-invariant measure on the manifold such that the action is measurably isomorphic to a relatively measure-preserving action over a standard boundary action.

Patterson–Sullivan theory for Anosov subgroups

We extend several notions and results from the classical Patterson–Sullivan theory to the setting of Anosov subgroups of higher rank semisimple Lie groups, working primarily with invariant Finsler metrics on associated symmetric spaces. In particular, we prove the equality between the Hausdorff dimensions of flag limit sets, computed with respect to a suitable Gromov (pre-)metric on the flag manifold, and the Finsler critical exponents of Anosov subgroups.

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.

Filling functions of arithmetic groups

The Dehn function and its higher-dimensional generalizations measure the difficulty of filling a sphere in a space by a ball. In nonpositively curved spaces, one can construct fillings using geodesics, but fillings become more complicated in subsets of nonpositively curved spaces, such as lattices in symmetric spaces. In this paper, we prove sharp filling inequalities for (arithmetic) lattices in higher rank semisimple Lie groups. When n is less than the rank of the associated symmetric space, we show that the n-dimensional filling volume function of the lattice grows at the same rate as that of the associated symmetric space, and when n is equal to the rank, we show that the n-dimensional filling volume function grows exponentially. This broadly generalizes a theorem of Lubotzky-Mozes-Raghunathan on length distortion in lattices and confirms conjectures of Thurston, Gromov, and Bux-Wortman.

Primitive stable representations in higher rank semisimple Lie groups

We study primitive stable representations of free groups into higher rank semisimple Lie groups and their properties. Let $$\Sigma $$ be a compact, connected, orientable surface (possibly with boundary) of negative Euler characteristic. We first verify $$\sigma _{mod}$$ -regularity for convex projective structures and positive representations. Then we show that the holonomies of convex projective structures and positive representations on $$\Sigma $$ are all primitive stable if $$\Sigma $$ has one boundary component.

Entropy, Lyapunov exponents, and rigidity of group actions

This text is an expanded series of lecture notes based on a 5-hour course given at the workshop entitled Workshop for young researchers: Groups acting on manifolds held in Teresopolis, Brazil in June 2016. The course introduced a number of classical tools in smooth ergodic theory-particularly Lya-punov exponents and metric entropy-as tools to study rigidity properties of group actions on manifolds. We do not present comprehensive treatment of group actions or general rigidity programs. Rather, we focus on two rigidity results in higher-rank dynamics: the measure rigidity theorem for affine Anosov abelian actions on tori due to A. Katok and R. Spatzier [114] and recent the work of the author with D. Fisher, S. Hurtado, F. Rodriguez Hertz, and Z. Wang on actions of lattices in higher-rank semisimple Lie groups on manifolds [31, 36]. We give complete proofs of these results and present sufficient background in smooth ergodic theory needed for the proofs. A unifying theme in this text is the use of metric entropy and its relation to the geometry of conditional measures along foliations as a mechanism to verify invariance of measures. i

Smooth factors of projective actions of higher-rank lattices and rigidity

We study smooth factors of the standard actions of lattices in higher rank semisimple Lie groups on flag manifolds. Under a mild condition on existence of a single differentiable sink, we show that these factors are $C^{\infty}$-conjugate to the standard actions on flag manifolds.

Convex cocompactness in pseudo-Riemannian hyperbolic spaces

Anosov representations of word hyperbolic groups into higher-rank semisimple Lie groups are representations with finite kernel and discrete image that have strong analogies with convex cocompact representations into rank-one Lie groups. However, the most naive analogy fails: generically, Anosov representations do not act properly and cocompactly on a convex set in the associated Riemannian symmetric space. We study representations into projective indefinite orthogonal groups $$\mathrm {PO}(p,q)$$ by considering their action on the associated pseudo-Riemannian hyperbolic space $$\mathbb {H}^{p,q-1}$$ in place of the Riemannian symmetric space. Following work of Barbot and Merigot in anti-de Sitter geometry, we find an intimate connection between Anosov representations and a natural notion of convex cocompactness in this setting.

Anosov subgroups: dynamical and geometric characterizations

We study infinite covolume discrete subgroups of higher rank semisimple Lie groups, motivated by understanding basic properties of Anosov subgroups from various viewpoints (geometric, coarse geometric and dynamical). The class of Anosov subgroups constitutes a natural generalization of convex cocompact subgroups of rank one Lie groups to higher rank. Our main goal is to give several new equivalent characterizations for this important class of discrete subgroups. Our characterizations capture “rank one behavior” of Anosov subgroups and are direct generalizations of rank one equivalents to convex cocompactness. Along the way, we considerably simplify the original definition, avoiding the geodesic flow. We also show that the Anosov condition can be relaxed further by requiring only non-uniform unbounded expansion along the (quasi)geodesics in the group.

Prime II1 factors arising from irreducible lattices in products of rank one simple Lie groups

Abstract We prove that if Γ is an icc irreducible lattice in a product of connected non-compact rank one simple Lie groups with finite center, then the II 1 {\mathrm{II}_{1}} factor L ⁢ ( Γ ) {L(\Gamma)} is prime. In particular, we deduce that the II 1 {\mathrm{II}_{1}} factors associated to the arithmetic groups PSL 2 ⁢ ( ℤ ⁢ [ d ] ) {\mathrm{PSL}_{2}(\mathbb{Z}[\sqrt{d}])} and PSL 2 ⁢ ( ℤ ⁢ [ S - 1 ] ) {\mathrm{PSL}_{2}(\mathbb{Z}[S^{-1}])} are prime for any square-free integer d ≥ 2 {d\geq 2} with d ≢ 1 ⁢ ( mod ⁡ 4 ) {d\not\equiv 1~{}(\operatorname{mod}\,4)} and any finite non-empty set of primes S. This provides the first examples of prime II 1 {\mathrm{II}_{1}} factors arising from lattices in higher rank semisimple Lie groups. More generally, we describe all tensor product decompositions of L ⁢ ( Γ ) {L(\Gamma)} for icc countable groups Γ that are measure equivalent to a product of non-elementary hyperbolic groups. In particular, we show that L ⁢ ( Γ ) {L(\Gamma)} is prime, unless Γ is a product of infinite groups, in which case we prove a unique prime factorization result for L ⁢ ( Γ ) {L(\Gamma)} .

Stabilizer rigidity in irreducible group actions

We consider irreducible actions of locally compact product groups, and of higher rank semi-simple Lie groups. Using the intermediate factor theorems of Bader-Shalom and Nevo-Zimmer, we show that the action stabilizers, and all irreducible invariant random subgroups, are co-amenable in their normal closure. As a consequence, we derive rigidity results on irreducible actions that generalize and strengthen the results of Bader-Shalom and Stuck-Zimmer.

The Nevo–Zimmer intermediate factor theorem over local fields

The Nevo–Zimmer theorem classifies the possible intermediate G-factors Y in , where G is a higher rank semisimple Lie group, P a minimal parabolic and X an irreducible G-space with an invariant probability measure. An important corollary is the Stuck–Zimmer theorem, which states that a faithful irreducible action of a higher rank Kazhdan semisimple Lie group with an invariant probability measure is either transitive or free, up to a null set. We present a different proof of the first theorem, that allows us to extend these two well-known theorems to linear groups over arbitrary local fields.

SOME RECENT RESULTS ON ANOSOV REPRESENTATIONS

In this note we give an overview of some of our recent work on Anosov representations of discrete groups into higher rank semisimple Lie groups.

Invariant random subgroups of the free group

Let G be a locally compact group. A random closed subgroup with conjugation-invariant law is called an invariant random subgroup or IRS for short. We show that each nonabelian free group has a large “zoo” of IRS’s. This contrasts with results of Stuck and Zimmer which show that there are no non-obvious IRS’s of higher rank semisimple Lie groups with property (T).

The extremal symmetry of arithmetic simplicial complexes

Let G be a higher-rank semisimple Lie group over a nonarchimedean local field, for example G=PGL(n,QP). To any lattice L in G there is an associated simplicial complex BL, given by the quotient by L of the Bruhat–Tits building associated to G. In this paper prove that the simplicial structure BL exhibits some remarkable and extremal symmetry properties, in particular when compared to any other simplicial structure on (any cover of) BL.

Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity

We study the geometry of non-relatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with non-relatively hyperbolic peripheral subgroups is a quasi-isometry invariant. As an application, Artin groups are relatively hyperbolic if and only if freely decomposable. We also introduce a new quasi-isometry invariant of metric spaces called metrically thick, which is sufficient for a metric space to be non-hyperbolic relative to any non-trivial collection of subsets. Thick finitely generated groups include: mapping class groups of most surfaces; outer automorphism groups of most free groups; certain Artin groups; and others. Non-uniform lattices in higher rank semisimple Lie groups are thick and hence non-relatively hyperbolic, in contrast with rank one which provided the motivating examples of relatively hyperbolic groups. Mapping class groups are the first examples of non-relatively hyperbolic groups having cut points in any asymptotic cone, resolving several questions of Drutu and Sapir about the structure of relatively hyperbolic groups. Outside of group theory, Teichmüller spaces for surfaces of sufficiently large complexity are thick with respect to the Weil–Peterson metric, in contrast with Brock–Farb’s hyperbolicity result in low complexity.

Groupes Convexes Cocompacts En Rang Supérieur

For higher rank semisimple Lie groups, we give an obstruction to the generalization of the notion of a discrete convex cocompact subgroup

Almost isometric actions, property (T), and local rigidity

Let Γ be a discrete group with property (T) of Kazhdan. We prove that any Riemannian isometric action of Γ on a compact manifold X is locally rigid. We also prove a more general foliated version of this result. The foliated result is used in our proof of local rigidity for standard actions of higher rank semisimple Lie groups and their lattices in [FM2].

Counting primes, groups, and manifolds.

Let Lambda=SL(2)(Z) be the modular group and let c(n)(Lambda) be the number of congruence subgroups of Lambda of index at most n. We prove that lim(n--> infinity )(log c(n)(Lambda)/((log n)(2)/log log n))=(3-2(sqrt)2)/4. The proof is based on the Bombieri-Vinogradov "Riemann hypothesis on the average" and on the solution of a new type of extremal problem in combinatorial number theory. Similar surprisingly sharp estimates are obtained for the subgroup growth of lattices in higher rank semisimple Lie groups. If G is such a Lie group and Gamma is an irreducible lattice of G it turns out that the subgroup growth of Gamma is independent of the lattice and depends only on the Lie type of the direct factors of G. It can be calculated easily from the root system. The most general case of this result relies on the Generalized Riemann Hypothesis, but many special cases are unconditional. The proofs use techniques from number theory, algebraic groups, finite group theory, and combinatorics.