Average bending of convex pleated planes in hyperbolic three-space

If P is a pleated plane in 3-dimensional hyperbolic space H ³ and α a geodesic in its intrinsic metric we define B(P,α), the average bending of P in the direction α. We show that if P is a convex pleated plane embedded in H ³ then B(P,α)≤K for some universal K. Furthermore if P_Γ is a boundary component of the convex hull of a quasi-Fuchsian group Γ then B(P_Γ,α)=B(Γ) almost everywhere, where B(Γ) is a constant times the length of the bending lamination β_Γ of the pleated surface X _Γ=P_Γ/Γ. We use these to prove a number of results about quasi-Fuchsian groups including a universal bound on the Lipschitz constant for the map to infinity and a bound on the length of β_Γ by a constant times the Euler characteristic of X _Γ. Oblatum 10-X-1996 & 23-V-1997     

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)). We also show thatG acts trivially on the coherent cohomology groupsH ⁱ(Γ/G, O) for anyi≥0.     

Structure of the automorphism group for partially commutative class two nilpotent groups

Let R be a ring, which is either a ring of integers or a field of zero characteristic. For every finite graph Γ, we construct an R-arithmetic linear group H(Γ). The group H(Γ) is realized as the factor automorphism group of a partially commutative class two nilpotent R-group G _Γ. Also we describe the structure of the entire automorphism group of a partially commutative nilpotent R-group of class two.     

On the analogue of Weil’s converse theorem for Jacobi forms and their lift to half-integral weight modular forms

We generalize Weil’s converse theorem to Jacobi cusp forms of weight k, index m and Dirichlet character χ over the group Γ ₀(N)⋉ℤ². Then two applications of this result are given; we generalize a construction of Jacobi forms due to Skogman and present a new proof for several known lifts of such Jacobi forms to half-integral weight modular forms.     

Critical exponents for groups of isometries

Let Γ be a convex co-compact group of isometries of a CAT(−1) space X and let Γ₀ be a normal subgroup of Γ. We show that, provided Γ is a free group, a sufficient condition for Γ and Γ₀ to have the same critical exponent is that Γ / Γ₀ is amenable.      

Commensurations of Out(F<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>)

Let Out(F _n ) denote the outer automorphism group of the free group F _n with n>3. We prove that for any finite index subgroup Γ<Out(F _n ), the group Aut(Γ) is isomorphic to the normalizer of Γ in Out(F _n ). We prove that Γ is co-Hopfian: every injective homomorphism Γ→Γ is surjective. Finally, we prove that the abstract commensurator Comm(Out(F _n )) is isomorphic to Out(F _n ).     

Automorphisms of groups and of schemes of finite type

We show first that certain automorphism groups of algebraic varieties, and even schemes, are residually finite and virtually torsion free. (A group virtually has a property if some subgroup of finite index has it.) The rest of the paper is devoted to a study of the groups of automorphisms. Aut(Γ) and outer automorphisms Out(Γ) of a finitely generated group Γ, by using the finite-dimensional representations of Γ. This is an old idea (cf. the discussion of Magnus in [11]). In particular the classes of semi-simplen-dimensional representations of Γ are parametrized by an algebraic varietyS _n (Γ) on which Out(Γ) acts. We can apply the above results to this action and sometimes conclude that Out(Γ) is residually finite and virtually torsion free. This is true, for example, when Γ is a free group, or a surface group. In the latter case Out(Γ) is a “mapping class group.” Partially supported by the NSF under Grant MCS 80-05802.     

Un critère de rationalité provenant de la géométrie non commutative

We prove a conjecture of A. Connes, which gives a rationality criterion for elements of the closure of ℂΓ (Γ a free group) in the space of bounded operators in l ² (Γ). We show that this criterion applies also to the ring of Malcev–Neumann series on Γ.

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Oblatum 2-VIII-1996 & 20-IX-1996     

The subgroup growth spectrum of virtually free groups

For a finitely generated group Γ denote by μ(Γ) the growth coefficient of Γ, that is, the infimum over all real numbers d such that s _n (Γ) < n! ^d . We show that the growth coefficient of a virtually free group is always rational, and that every rational number occurs as growth coefficient of some virtually free group.     

Séries de poincaré des groupes géométriquement finis

In this paper, we study the behaviour of the Poincaré series of a geometrically finite group Γ of isometries of a riemannian manifoldX with pinched curvature, in the case when Γ contains parabolic elements. We give a sufficient condition on the parabolic subgroups of Γ in order that Γ be of divergent type. When Γ is of divergent type, we show that the Sullivan measure on the unit tangent bundle ofX/Γ is finite if and only if certain series which involve only parabolic elements of Γ are convergent. We build also examples of manifoldsX on which geometrically finite groups of convergent type act.

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Durant la rédaction de cet article, M. Peigné a bénéficié d'un détachement au Centre National de la Recherche Scientifique, URA 305.     

Discrete subgroups of algebraic groups over local fields of positive characteristics

It is shown in this paper that ifG is the group ofk-points of a semisimple algebraic groupG over a local fieldk of positive characteristic such that all itsk-simple factors are ofk-rank 1 and Γ ⊂G is a non-cocompact irreducible lattice then Γ admits a fundamental domain which is a union of translates of Siegel domains. As a consequence we deduce that ifG has more than one simple factor, then Γ is finitely generated and by a theorem due to Venkataramana, it is arithmetic.     

Symbolic dynamics for triangle groups

We generalize E. Artin’s continued fraction coding of the geodesics on the modular surface to any finite index subgroup Θ of a nonuniform hyperbolic triangle group Γ. D. Mayer’s study of the Selberg zeta function of PSl (2, Z ) is extended to Θ and its group representations. We give representatives for Γ-primitive conjugacy classes and derive a Markov system of interval maps for Γ and a Markov partition for the billiard flow on Γ\ SH ² . This leads to identities for values of the dilogarithm function at algebraic numbers. We also find the Γ-analogues of Gauss measure on [0,1]. Oblatum 16-VIII-1993 & 15-VIII-1994 & 2-I-1996     

Fatou–Bieberbach Domains as Basins of Attraction of Automorphisms Tangent to the Identity

We prove that there exists an automorphism of ℂ² tangent to the identity with a domain of attraction D to the origin, biholomorphic to ℂ², along a degenerate characteristic direction. Our automorphism of ℂ² is conjugate to a translation in D. We also prove the existence of a curve Γ, a biholomorphic copy of ℂ, entirely contained in the boundary of D. In our construction Γ is tangent to the z-axis in a neighborhood of the origin. The automorphisms we construct also fix the w-axis; therefore we obtain D, a Fatou–Bieberbach domain that does not intersect two biholomorphic copies of ℂ locally transversal at the origin.     

Embedding of hyperbolic groups into products of binary trees

We show that every Gromov hyperbolic group Γ admits a quasi-isometric embedding into the product of n+1 binary trees, where n=dim∂_∞Γ is the topological dimension of the boundary at infinity of Γ.     

On graphs related to conjugacy classes of groups

LetG be a finite group. Attach toG the following two graphs: Γ — its vertices are the non-central conjugacy classes ofG, and two vertices are connected if their sizes arenot coprime, and Γ* — its vertices are the prime divisors of sizes of conjugacy classes ofG, and two vertices are connected if they both divide the size of some conjugacy class ofG. We prove that whenever Γ* is connected then its diameter is at most 3, (this result was independently proved in [3], for solvable groups) and Γ* is disconnected if and only ifG is quasi-Frobenius with abelian kernel and complements. Using the method of that proof we give an alternative proof to Theorems in [1],[2],[6], namely that the diameter of Γ is also at most 3, whenever the graph is connected, and that Γ is disconnected if and only ifG is quasi-Frobenius with abelian kernel and complements. As a result we conclude that both Γ and Γ* have at most two connected components. In [2],[3] it is shown that the above bounds are best possible. The content of this paper corresponds to a part of the author’s Ph.D. thesis carried out at the Tel Aviv University under the supervision of Prof. Marcel Herzog.     

A lattice in more than two Kac-Moody groups is arithmetic

Let Γ < G ₁ × … × G _n be an irreducible lattice in a product of infinite irreducible complete Kac-Moody groups of simply laced type over finite fields. We show that if n ≥ 3, then each G _i is a simple algebraic group over a local field and Γ is an S-arithmetic lattice. This relies on the following alternative which is satisfied by any irreducible lattice provided n ≥ 2: either Γ is an S-arithmetic (hence linear) group, or Γ is not residually finite. In that case, it is even virtually simple when the ground field is large enough. More general CAT(0) groups are also considered throughout.     

Free group representations from vector-valued multiplicative functions,I

Let Γ denote a noncommutative free group and let Ω stand for its boundary. We construct a large class of unitary representations of Γ. This class contains many previously studied representations, and is closed under several natural operations. Each of the constructed representations is in fact a representation of Γ ⋉_λ C(Ω). We prove here that each of them is irreducible as a representation of Γ ⋉_λ C(Ω). Actually, as will be shown in further work, each of them is irreducible as a representation of Γ, or is the direct sum of exactly two irreducible, inequivalent Γ-representations. This research was supported by the Italian CNR.