Non-elementary Subgroups Research Articles

Lower bound on growth of non-elementary subgroups in relatively hyperbolic groups

Abstract This paper proves that, in a non-elementary relatively hyperbolic group, the logarithm growth rate of any non-elementary subgroup has a linear lower bound by the logarithm of the size of the corresponding generating set. As a consequence, any non-elementary subgroup has uniform exponential growth.

Statistical Hyperbolicity for Harmonic Measure

Abstract We consider harmonic measures that arise from random walks on the mapping class group determined by probability distributions that have finite first moments with respect to the Teichmüller metric and whose supports generate nonelementary subgroups. We prove that Teichmüller space with the Teichmüller metric is statistically hyperbolic for such a harmonic measure.

Intersecting limit sets of Kleinian subgroups and Susskind’s question

We construct a non-elementary Fuchsian group that admits two non-elementary subgroups with trivial intersection and whose radial limit sets intersect non-trivially. This negatively answers a question of Perry Susskind (1989) that was stated as a conjecture by James W. Anderson (2014).

A classification ofℝ-Fuchsian subgroups of Picard modular groups

Given an imaginary quadratic extension K of ℚ, we classify the maximal nonelementary subgroups of the Picard modular group PU(1,2;𝒪 K ) preserving a totally real totally geodesic plane in the complex hyperbolic plane ℍ ℂ 2 . We prove that these maximal ℝ-Fuchsian subgroups are arithmetic, and describe the quaternion algebras from which they arise. For instance, if the radius Δ of the corresponding ℝ-circle lies in ℕ-{0}, then the stabiliser arises from the quaternion algebra Δ,|D K | ℚ . We thus prove the existence of infinitely many orbits of K-arithmetic ℝ-circles in the hypersphere of ℙ 2 (ℂ).

A classification of $\mathbb{C}$-Fuchsian subgroups of Picard modular groups

Given an imaginary quadratic extension $K$ of $\mathbb{Q}$, we give a classification of the maximal nonelementary subgroups of the Picard modular group $\operatorname{PSU}_{1,2}(\mathcal{O}_K)$ preserving a complex geodesic in the complex hyperbolic plane $\mathbb{H}^2_\mathbb{C}$. Complementing work of Holzapfel, Chinburg-Stover and M\"oller-Toledo, we show that these maximal $\mathbb{C}$-Fuchsian subgroups are arithmetic, arising from a quaternion algebra $\Big(\!\begin{array}{c} D\,,D_K\\\hline\mathbb{Q}\end{array} \!\Big)$ for some explicit $D\in\mathbb{N}-\{0\}$ and $D_K$ the discriminant of $K$. We thus prove the existence of infinitely many orbits of $K$-arithmetic chains in the hypersphere of $\mathbb{P}_2(\mathbb{C})$.

Expansion, random walks and sieving in $$S{L_2}({\mathbb{F}_p}[t])$$

We construct new examples of expander Cayley graphs of finite groups, arising as congruence quotients of non-elementary subgroups of \(S{L_2}({\mathbb{F}_p}[t])\) modulo certain square-free ideals. We describe some applications of our results to simple random walks on such subgroups, specifically giving bounds on the rate of escape of such walks from algebraic subvarieties, the set of squares and the set of elements with reducible characteristic polynomial in \(S{L_2}({\mathbb{F}_p}[t])\).

Discreteness is undecidable

We prove that the discreteness problem for two-generated nonelementary subgroups of [Formula: see text] is undecidable in the Blum–Shub–Smale (BSS) computability model.

On geometric convergence of discrete groups

One of the basic questions in the Kleinian group theory is to understand both algebraic and geometric limiting behavior of sequences of discrete subgroups. In this paper we consider the geometric convergence in the setting of the isometric group of the real or complex hyperbolic space. It is known that if Γ is a non-elementary finitely generated group and ϱ i : Γ → SO(n, 1) a sequence of discrete and faithful representations, then the geometric limit of ϱ i (Γ) is a discrete subgroup of SO(n, 1). We generalize this result by showing that for a sequence of discrete and non-elementary subgroups {G j } of SO(n, 1) or PU(n, 1), if {G j } has uniformly bounded torsion, then its geometric limit is either elementary, or discrete and non-elementary.

Discreteness criteria in PU (1, n; C)

In this paper, we study the discreteness for non-elementary subgroups in PU (1, n; C), and several discreteness criteria are obtained.

MINIMA OF BINARY INDEFINITE HERMITIAN FORMS

Classification of binary indefinite primitive Hermitian forms modulo the action of the extended Bianchi group (or Hilbert modular group) Bd is given. When the discriminant of the quadratic field [Formula: see text] (and d) is negative, the results obtained can be applied to classify the maximal non-elementary Fuchsian subgroups of Bd, and to find the Hermitian points in the Markov spectrum of Bd. If ν is a Hermitian point in the spectrum, then there is a set of extremal geodesics in H3, the upper half-space model of the three-dimensional hyperbolic space, with diameter 1/ν, which depends on one continuous parameter.

DISCRETE AND DENSE SUBGROUPS ACTING ON COMPLEX HYPERBOLIC SPACE

AbstractIn this paper, we study the discreteness criteria for nonelementary subgroups of U(1,n;ℂ) acting on complex hyperbolic space. Several discreteness criteria are obtained. As applications, we obtain a classification of nonelementary subgroups of U(1,n;ℂ) and show that any dense subgroup of SU(1,n;ℂ) contains a dense subgroup generated by at most n elements when n≥2. We also obtain a necessary and sufficient condition for the normalizer of a discrete and nonelementary subgroup in SU(1,n;ℂ) to be discrete.

Discreteness criteria and algebraic convergence theorem for subgroups in $\mathit{PU}(1,n;C)$

In this paper, we will study the discreteness criterion for non-elementary subgroups in $\mathit{PU}(1,n;C)$. Several discreteness criteria are obtained. As an application, the convergence theorem of discrete subgroups in $\mathit{PU}(1,n;C)$ is discussed.

Ideal bicombings for hyperbolic groups and applications

For every hyperbolic group and more general hyperbolic graphs, we construct an equivariant ideal bicombing: this is a homological analogue of the geodesic flow on negatively curved manifolds. We then construct a cohomological invariant which implies that several Measure Equivalence and Orbit Equivalence rigidity results established in Monod and Shalom (Orbit equivalence rigidity and bounded cohomology, preprint, to appear) hold for all non-elementary hyperbolic groups and their non-elementary subgroups. We also derive superrigidity results for actions of general irreducible lattices on a large class of hyperbolic metric spaces.

Une alternative sur l'entropie des groupes

We prove the following alternative: either there exists a finitely generated group with exponential growth whose entropy is zero, or there exists a universal constant M>0 such that the entropy of all non-elementary hyperbolic groups with cyclic centralizers and their non-elementary subgroups is at least M. To cite this article: V. Guirardel, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 743–746.

The Poisson boundary of the mapping class group

A theory of random walks on the mapping class group and its non-elementary subgroups is developed. We prove convergence of sample paths in the Thurston compactification and show that the space of projective measured foliations with the corresponding harmonic measure can be identified with the Poisson boundary of random walks. The methods are based on an analysis of the asymptotic geometry of Teichmuller space and of the contraction properties of the action of the mapping class group on the Thurston boundary. We prove, in particular, that Teichmuller space is roughly isometric to a graph with uniformly bounded vertex degrees. Using our analysis of the mapping class group action on the Thurston boundary we prove that no non-elementary subgroup of the mapping class group can be a lattice in a higher rank semi-simple Lie group.