Counting and equidistribution over primes in hyperbolic groups

We study the growth of double cosets in the class of groups with contracting elements, including relatively hyperbolic groups, CAT(0) groups and mapping class groups among others. Generalizing a recent work of Gitik and Rips about hyperbolic groups, we prove that the double coset growth of two Morse subgroups of infinite index is comparable with the orbital growth function. The same result is further obtained for a more general class of subgroups whose limit sets are proper subsets in the entire limit set of the ambient group. The limit sets under consideration are defined in a general convergence compactification, including Gromov boundary, Bowditch boundary, Thurston boundary and horofunction boundary. As an application, we confirm a conjecture of Maher that hyperbolic 3‐manifolds are exponentially generic in the set of 3‐manifolds built from Heegaard splitting using complexity in Teichmüller metric.

Using authors's methods of 1980Using authors's methods of , 1981, some explicit finite sets of number fields containing all ground fields of arithmetic hyperbolic reflection groups in dimension at least 3 are defined, and explicit bounds of their degrees (over Q) are obtained.Thus, now, explicit bound of degree of ground fields of arithmetic hyperbolic reflection groups is known in all dimensions.Thus, now, we can, in principle, obtain effective finite classification of arithmetic hyperbolic reflection groups in all dimensions together.

Double Coset Decompositions of Groups

The paper continues from the work of Nikulin. Using our methods of 1980 and 1981, we define some explicit finite sets of number fields containing all ground fields of arithmetic hyperbolic reflection groups in dimensions at least 3, and we give good upper bounds for their degrees (over ℚ). This extends the earlier results of Nikulin for dimensions at least 4. This finally delivers a possibility, in principle, of effective finite classification of maximal arithmetic hyperbolic reflection groups (more generally, of reflective hyperbolic lattices) in all dimensions. Our results also give another proof of finiteness in dimension 3. In fact, using our methods, we show that finiteness in dimension 3 follows from finiteness in dimension 2.

Here we show that if Γ is an arithmetic Fuchsian group of genus 0, then the totally real defining field k of Γ must be such that [k : ℚ] ⩽ 11. The same inequality holds for discrete arithmetic hyperbolic reflection groups acting on a two-dimensional hyperbolic space ℍ2. In addition, we show that there exists an arithmetic Fuchsian group of genus 0 containing an element of order N if and only if N ∈ {2, 3, …, 16, 18, 20, 22, 24, 26, 28, 30, 36}. A slightly less precise statement holds for discrete arithmetic hyperbolic reflection groups acting on ℍ2.

We show that if H H is a non-elementary hyperbolic commensurated subgroup of infinite index in a hyperbolic group G G , then H H is virtually a free product of hyperbolic surface groups and free groups. We prove that whenever a one-ended hyperbolic group H H is a fiber of a non-trivial hyperbolic bundle then H H virtually splits over a 2-ended subgroup.

Geometry and arithmetic cycles attached to [formula omitted]—I

ABSTRACTLet N be a positive square-free integer such that the discrete group Γ0(N)+ has genus one. In a previous article, we constructed canonical generators xN and yN of the holomorphic function field associated with Γ0(N)+ as well as an algebraic equation PN(xN, yN) = 0 with integer coefficients satisfied by these generators. In the present paper, we study the singular moduli problem corresponding to xN and yN, by which we mean the arithmetic nature of the numbers xN(τ) and yN(τ) for any CM point τ in the upper half plane . If τ is any CM point which is not equivalent to an elliptic point of Γ0(N)+, we prove that the complex numbers xN(τ) and yN(τ) are algebraic integers. Going further, we characterize the algebraic nature of xN(τ) as the generator of a certain ring class field of of prescribed order and discriminant depending on properties of τ and level N. The theoretical considerations are supplemented by computational examples. As a result, several explicit evaluations are given for various N and τ, and further arithmetic consequences of our analysis are presented. In one example, we explicitly construct a set of minimal polynomials for the Hilbert class field of whose coefficients are less than 2.2 × 104, whereas the minimal polynomial obtained from the Hauptmodul of have coefficients as large as 6.6 × 1073.

In this chapter, we discuss similarities, differences and interaction between two natural and important classes of groups: arithmetic subgroups $\Gamma$ of Lie groups $G$ and mapping class groups mod of surfaces of genus $g$ with $n$ punctures. We also mention similar properties and problems for related groups such as outer automorphism groups $\operatorname{Out}(F\_n)$, Coxeter groups and hyperbolic groups. Since groups are often effectively studied by suitable spaces on which they act, we also discuss related properties of actions of arithmetic groups on symmetric spaces and actions of mapping class groups on Teichmüller spaces. Interaction between locally symmetric spaces and moduli spaces of Riemann surfaces through the example of the Jacobian map will also be discussed in the last part of this chapter. Since reduction theory, i.e., finding good fundamental domains for proper actions of discrete groups, is crucial to transformation group theory, i.e., to understand the algebraic structures of groups, properties of group actions and geometry, topology and compactifications of the quotient spaces, we discuss many different approaches to reduction theory of arithmetic groups acting on symmetric spaces. These results for arithmetic groups motivate some results on fundamental domains for the action of mapping class groups on Teichmüller spaces. For example, the Minkowski reduction theory of quadratic forms is generalized to the action of $\operatorname{Mod}g=\operatorname{Mod}{g, 0}$ on the Teichmüller space $\mathcal{T}\_g$ to construct an intrinsic fundamental domain consisting of finitely many cells, solving a weaker version of a folklore conjecture in the theory of Teichmüller spaces.

CHAPTER 5 - The Poincaré Model

This paper is devoted to some counting functions of level one and level three in the case of quotient space generated by some strictly hyperbolic Fuchsian group and the upper half-plane. Each of the functions is represented as a sum of some explicit part plus the error term. The explicit part is indexed over singularities of the corresponding Selberg zeta function. In particular, the obtained error term is not larger than O ( x 3/4) . The method applied in this paper follows traditional approach for achieving the error terms in the case of locally symmetric spaces of real rank one. In order to establish an analogy with the classical case, we consider the counting functions divided by x and x3, respectively.

We prove that there are only finitely many conjugacy classes of arithmetic maximal hyperbolic reflection groups.

This paper is a follow-up to the paper I. Agol, M. Belolipetsky, P. Storm, K. Whyte, Finiteness of arithmetic hyperbolic reflection groups, Groups, Geometry, and Dynamics 2 (2008), 481–498. The main purpose is to investigate the effective side of the method developed there and its possible application to the problem of classification of arithmetic hyperbolic reection groups.

We classify finite order symmetries g of the 14 exceptional unimodal function singularities f in 3 variables, which satisfy a so-called splitting condition. This means that the rank 2 positive subspace in the vanishing homology of f should not be contained in one eigenspace of g?. We also obtain a description of the hyperbolic complex reflection groups appearing as equivariant monodromy groups acting on the hyperbolic eigensubspaces arising. One of the most famous classical results in singularity theory is the Arnold and Brieskorn discovery of the close relationship between simple function singularities and Weyl groups Aμ, Dμ, Eμ [1, 6]. A few years after it, Arnold extended the relationship to simple singularities with the Z2 reflection symmetry and Weyl groups Bμ, Cμ, F4 [2] (see also Slodowy’s book [24]). Consideration of Zm symmetries of simple functions led in [10, 11, 12, 25] to the appearance of Shephard-Todd groups within function singularities. The emphasis there was on realisations of the complex reflection groups as equivariant monodromy groups acting on the appropriate character subspaces in the homology of invariant Milnor fibres, and on the diffeomorphisms between the discriminants of the reflection groups and of the Zm-equivariant functions. A further series of papers [13, 14, 15], on cyclic symmetries of the parabolic functions, brought in similar singularity realisations of certain complex crystallographic groups [22]. In this paper, we are naturally expanding the programme to cyclic symmetries of the 14 exceptional unimodal function singularities on one hand, and complex hyperbolic reflection groups on the other. The basic idea is as follows. In the 3-variable case, the intersection form on the vanishing homology of an exceptional unimodal function f is non-degenerate and has positive signature 2. Assume g is an automorphism of C of finite order m, and our function is g-invariant. Then g acts on the second homology of the Milnor fibre f−1(e), and decomposes it into a direct sum of the character subspaces Hχ, χ = 1, on which g acts as multiplication by χ. Assume the rank 2 positive subspace of the intersection form splits between two character summands. Then the monodromy within a g-invariant versal deformation of f acts as a complex hyperbolic reflection group on each of them. Developing further the technique introduced in papers on cyclically symmetric functions [10, 11, 12], we construct vanishing bases in the hyperbolic summands and obtain the generating reflections as the corresponding Picard-Lefschetz operators. The main result of the paper is a complete classification of the invariant symmetries of the 14 singularities, which split the positive subspace in the vanishing homology, and the description – via constructing the corresponding Dynkin diagrams – of the complex hyperbolic groups arising. All the rank 2 reflection groups obtained projectivise to the triangle groups of the Poincare disk. The task of identification of higher dimensional groups is left for a future paper, along with the consideration of the equivariant symmetry setting. It should be noted that it is the first time when complex hyperbolic reflection groups are appearing in a singularity theory context. The approach introduced may be useful for constructing new complex hyperbolic lattices (cf. [8, 20]). The paper is organised as follows. Section 1 introduces the notion of singularities with symmetry, recalls the definitions and constructions given in [10, 11, 12]. Section 2 contains classification of splitting invariant symmetries of the 14 singularities. In Section 3.3 we construct Dynkin diagrams of the hyperbolic monodromy groups associated with the symmetric functions. Projectivisations of the rank 2 monodromy groups are considered in Section 4. More details of the constructions may be found in [16].

In this paper, we determine the q-expansions of vector-valued modular forms (Knopp and Mason in Ill. J. Math. 48:1345–1366, 2004; Acta Arith. 110(2): 117–124, 2003) of large negative weight on the full modular group where we allow poles in the upper half plane and at infinity.