Finiteness of arithmetic hyperbolic reflection groups

A new proof of finiteness of maximal arithmetic reflection groups

We give a new proof of the finiteness of maximal arithmetic reflection groups. Our proof is novel in that it makes no use of trace formulas or other tools from the theory of automorphic forms and instead relies on the arithmetic Margulis lemma of Fraczyk, Hurtado and Raimbault.

Finiteness of arithmetic Kleinian reflection groups

We prove that there are only finitely many arithmetic Kleinian maximal reflection groups.

Higher finiteness properties of arithmetic approximate lattices: The Rank Theorem for number fields

We introduce geometric and homological finiteness properties for countable approximate groups via coarse geometry and then study these finiteness properties for S-arithmetic reductive approximate groups. For S-arithmetic approximate groups without infinite places we show that the finiteness length is finite and compute this finiteness length explicitly. In the simple case it is one less than the sum of the local ranks. This extends the Rank Theorem of Bux, Köhl and the second author from positive characteristic to characteristic zero. Our proof is based on a geometric version of their proof, b…

Arithmetic group symmetry and finiteness properties of Torelli groups

We examine groups whose resonance varieties, characteristic varieties and Sigma-invariants have a natural arithmetic group symmetry, and we explore implications on various finiteness properties of subgroups. We compute resonance varieties, characteristic varieties and Alexander polynomials of Torelli groups, and we show that all subgroups containing the Johnson kernel have finite first Betti number, when the genus is at least four. We also prove that, in this range, the $I$-adic completion of the Alexander invariant is finite-dimensional, and the Kahler property for the Torelli group implies t…

On the geometry of global function fields, the Riemann-Roch theorem, and finiteness properties of S-arithmetic groups

Harder's reduction theory provides filtrations of euclidean buildings that allow one to deduce cohomological and homological properties of S-arithmetic groups over global function fields. In this survey I will sketch the main points of Harder's reduction theory starting from Weil's geometry of numbers and the Riemann-Roch theorem, describe a filtration that is particularly useful for deriving finiteness properties of S-arithmetic groups, and state the rank conjecture and its partial verifications that do not restrict the cardinality of the underlying field of constants. As a motivation for fur…

Finiteness of the number of arithmetic groups generated by reflections in Lobachevsky spaces

After results by the author (1980, 1981), and by Vinberg (1981), finiteness of the number of maximal arithmetic reflection groups in Lobachevsky spaces was not known in dimensions $2\le n\le 9$ only. Recently (2005), the finiteness was proved in dimension 2 by Long, Maclachlan and Reid, and in dimension 3 by Agol. Here we use these results in dimensions 2 and 3 to prove finiteness in all remaining dimensions $4\le n\le 9$. Methods of the author (1980, 1981) are strong enough to complete this in few lines by simple considerations.

Arithmeticity of complex hyperbolic triangle groups

Complex hyperbolic triangle groups were first considered by Mostow in building the first nonarithmetic lattices in PU(2, 1). They are a natural generalization of the classical triangle groups acting on the hyperbolic plane. A well-known theorem of Takeuchi is that there are only finitely many Fuchsian triangle groups that are also an arithmetic lattice in PSL_2(R). We consider similar finiteness theorem for complex hyperbolic triangle groups. In particular, we show that there are finitely many complex hyperbolic triangle groups with rational angular invariant which determine an arithmetic latt…

Arithmetic hyperbolic reflection groups

A hyperbolic reflection group is a discrete group generated by reflections in the faces of an $n$-dimensional hyperbolic polyhedron. This survey article is dedicated to the study of arithmetic hyperbolic reflection groups with an emphasis on the results that were obtained in the last ten years and on the open problems.

Effective virtual and residual properties of some arithmetic hyperbolic 3-manifolds

We give an effective upper bound, for certain arithmetic hyperbolic 3-manifold groups obtained from a quadratic form construction, on the minimal index of a subgroup that embeds in a fixed 6-dimensional right-angled reflection group, stabilizing a totally geodesic subspace. In particular, for manifold groups in any fixed commensurability class we show that the index of such a subgroup is asymptotically smaller than any fractional power of the volume of the manifold. We also give effective bounds on the geodesic residual finiteness growths of closed hyperbolic manifolds that totally geodesicall…

Connectivity Properties of Horospheres in Euclidean Buildings and Applications to Finiteness Properties of Discrete Groups

Let G(O_S) be an S-arithmetic subgroup of a connected, absolutely almost simple linear algebraic group G over a global function field K. We show that the sum of local ranks of G determines the homological finiteness properties of G(O_S) provided the K-rank of G is 1. This shows that the general upper bound for the finiteness length of G(O_S) established in an earlier paper is sharp in this case. The geometric analysis underlying our result determines the conectivity properties of horospheres in thick Euclidean buildings.

Effective bounds for Vinberg's algorithm for arithmetic hyperbolic lattices

A group of isometries of a hyperbolic $n$-space is called a reflection group if it is generated by reflections in hyperbolic hyperplanes. Vinberg gave a semi-algorithm for finding a maximal reflection sublattice in a given arithmetic subgroup of $O(n,1)$ of the simplest type. We provide an effective termination condition for Vinberg's semi-algorithm with which it becomes an algorithm for finding maximal reflection sublattices. The main new ingredient of the proof is an upper bound for the number of faces of an arithmetic hyperbolic Coxeter polyhedron in terms of its volume.

Finiteness properties for some rational Poincaré duality groups

A combination of Bestvina--Brady Morse theory and an acyclic reflection group trick produces a torsion-free finitely presented Q-Poincaré duality group which is not the fundamental group of an aspherical closed ANR Q-homology manifold. The acyclic construction suggests asking which Q-Poincaré duality groups act freely on Q-acyclic spaces, i.e., which groups are FH(Q). For example, the orbifold fundamental group Γ of a good orbifold satisfies Q-Poincaré duality, and we show Γ is FH(Q) if the Euler characteristics of certain fixed sets vanish.

双曲混合流形中极大测地子流形的有限性

Finiteness of Maximal Geodesic Submanifolds in Hyperbolic Hybrids

该研究证明大类非算术双曲n-流形（含Gromov等的混合流形）的有限体积浸入全测地超曲面有限，且高余维下维数≥2的极大测地子流形有限，证明结合算术群结构、动力系统与负曲率几何。

A finiteness theorem for hyperbolic 3-manifolds

We prove that there are only finitely many closed hyperbolic 3-manifolds with injectivity radius and first eigenvalue of the Laplacian bounded below whose fundamental groups can be generated by a given number of elements. An application to arithmetic manifolds is also given.

On the Residual Finiteness Growths of Particular Hyperbolic Manifold Groups

We give a quantification of residual finiteness for the fundamental groups of hyperbolic manifolds that admit a totally geodesic immersion to a compact, right-angled Coxeter orbifold of dimension 3 or 4. Specifically, we give explicit upper bounds on residual finiteness that are linear in terms of geodesic length. We then extend the linear upper bounds to hyperbolic manifolds with a finite cover that admits such an immersion. Since the quantifications are given in terms of geodesic length, we define the geodesic residual finiteness growth and show that this growth is equivalent to the usual re…

Infinitely many quasi-arithmetic maximal reflection groups

In contrast to the fact that there are only finitely many maximal arithmetic reflection groups acting on the hyperbolic space $\mathbb{H}^n$, $n\geq 2$, we show that: (a) one can produce infinitely many maximal quasi-arithmetic reflection groups acting on $\mathbb{H}^2$; (b) they admit infinitely many different fields of definition; (c) the degrees of their fields of definition are unbounded. However, for $n\geq 14$ an approach initially developed by Vinberg shows that there are still finitely many fields of definitions in the quasi-arithmetic case.

On fields of definition of arithmetic Kleinian reflection groups

We show that degrees of the real fields of definition of arithmetic Kleinian reflection groups are bounded by 35.

Dirichlet-Ford Domains and Arithmetic Reflection Groups

In this paper, it is shown that a Fuchsian group, acting on the upper half-plane model for $\mathbb{H}^2$, admits a Ford domain which is also a Dirichlet domain, for some center, if and only if it is an index 2 subgroup of a reflection group. This is used to exhibit an example of a maximal arithmetic hyperbolic reflection group which is not congruence. Analogous results, and counterexamples, are given in the case of Kleinian groups.

Thompson groups for systems of groups, and their finiteness properties

We describe a procedure for constructing a generalized Thompson group out of a family of groups that is equipped with what we call a cloning system. The previously known Thompson groups F, V, Vbr and Fbr arise from this procedure using, respectively, the systems of trivial groups, symmetric groups, braid groups and pure braid groups. We give new examples of families of groups that admit a cloning system and study how the finiteness properties of the resulting generalized Thompson group depend on those of the original groups. The main new examples here include upper triangular matrix groups, mo…

On ground fields of arithmetic hyperbolic reflection groups. III

This paper continues arXiv.org:math.AG/0609256, arXiv:0708.3991 and arXiv:0710.0162 . Using authors's methods of 1980, 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 togeth…

On ground fields of arithmetic hyperbolic reflection groups. II

This paper continues arXiv.org:math.AG/0609256 and arXiv:0708.3991 Using authors's methods of 1980, 1981, some explicit finite sets of number fields containing all ground fields of arithmetic hyperbolic reflection groups in dimensions at least 4 are defined, and good explicit bounds of their degrees (over Q) are obtained. This could be important for further classification. Thus, now, an explicit bound of degree of ground fields of arithmetic hyperbolic reflection groups is unknown in dimension 3 only.

On ground fields of arithmetic hyperbolic reflection groups

Using authors's methods of 1980, 1981, some explicit finite sets of number fields containing ground fields of arithmetic hyperbolic reflection groups are defined, and good bounds of their degrees (over Q) are obtained. For example, degree of the ground field of any arithmetic hyperbolic reflection group in dimension at least 6 is bounded by 56. These results could be important for further classification. We also formulate a mirror symmetric conjecture to finiteness of the number of arithmetic hyperbolic reflection groups which was established in full generality recently.

On fields of definition of arithmetic Kleinian reflection groups II

Following the previous work of Nikulin and Agol, Belolipetsky, Storm, and Whyte it is known that there exist only finitely many (totally real) number fields that can serve as fields of definition of arithmetic hyperbolic reflection groups. We prove a new bound on the degree $n_k$ of these fields in dimension 3: $n_k$ does not exceed 9. Combined with previous results of Maclachlan and Nikulin, this leads to a new bound $n_k \le 25$ which is valid for all dimensions. We also obtain upper bounds for the discriminants of these fields and give some heuristic results which may be useful for the clas…

全测地超曲面的有限性

Finiteness of totally geodesic hypersurfaces

本文研究闭负曲率解析黎曼流形中全测地超曲面的数量问题，证明其有限性，除非流形为常曲率（此时等距于算术双曲流形），揭示了全测地超曲面数量与流形曲率性质的深刻关联。

The transition constant for arithmetic hyperbolic reflection groups

The transition constant was introduced in our 1981 paper and denoted as N(14). It is equal to the maximal degree of the ground fields of V-arithmetic connected edge graphs with 4 vertices and of the minimality 14. This constant is fundamental since if the degree of the ground field of an arithmetic hyperbolic reflection group is greater than N(14), then the field comes from very special plane reflection groups. In our recent paper (see also arXiv:0708.3991), we claimed its upper bound 56. Using similar but more difficult considerations, here we improve this bound. These results could be import…

Some finiteness results for groups of automorphisms of manifolds

We prove that in dimensions not equal to 4, 5, or 7, the homology and homotopy groups of the classifying space of the topological group of diffeomorphisms of a disk fixing the boundary are finitely generated in each degree. The proof uses homological stability, embedding calculus and the arithmeticity of mapping class groups. From this we deduce similar results for the homeomorphisms of $R^n$ and various types of automorphisms of 2-connected manifolds.

Computing arithmetic invariants for hyperbolic reflection groups

We describe a collection of computer scripts written in PARI/GP to compute, for reflection groups determined by finite-volume polyhedra in $\mathbb{H}^3$, the commensurability invariants known as the invariant trace field and invariant quaternion algebra. Our scripts also allow one to determine arithmeticity of such groups and the isomorphism class of the invariant quaternion algebra by analyzing its ramification. We present many computed examples of these invariants. This is enough to show that most of the groups that we consider are pairwise incommensurable. For pairs of groups with identica…

Self-correspondences of K3 surfaces via moduli of sheaves and arithmetic hyperbolic reflection groups

An integral hyperbolic lattice is called reflective if its automorphism group is generated by reflections, up to finite index. Since 1981, it is known that their number is essentially finite. We show that K3 surfaces over C with reflective Picard lattices can be characterized in terms of compositions of their self-correspondences via moduli of sheaves with primitive isotropic Mukai vector: Their self-correspondences with integral action on the Picard lattice are numerically equivalent to compositions of a finite number of especially simple self-correspondences via moduli of sheaves. This relat…

Kleinian 球填充、反射群与算术性

Kleinian sphere packings, reflection groups, and arithmeticity

本文研究Kleinian球填充等相关问题，解决了高维晶体球填充存在性难题，提出几何加倍构造方法并探究真整填充，技术依托洛伦兹二次型等相关计算。

双曲群的子群、有限性性质与复双曲格

Subgroups of hyperbolic groups, finiteness properties and complex hyperbolic lattices

本文证明最简单类型余紧复双曲算术格的足够深有限指数子群存在到整数群的大量同态，其核具有类型F_{m-1}而非F_m，从而构造出双曲群的许多有限展示非双曲子群，回答了Brady的长期问题，并给出非球面Kähler流形Singer猜想的一个特殊情形证明。

Cantor流形的渐近映射类群及其有限性性质

Asymptotic mapping class groups of Cantor manifolds and their finiteness properties

该研究证明了特定曲面渐近映射类群的$F_\infty$型有限性，提出更一般的Cantor流形渐近映射类群$F_\infty$型判定定理，给出多类重要示例并验证了部分流形下其同调与稳定同调的一致性。

混合流形中超平面的定量有限性

Quantitative finiteness of hyperplanes in hybrid manifolds

本研究针对由Gromov与Piatetski-Shapiro粘合构造生成的n≥3维非算术双曲流形，证明其全测地超平面的定量有限性，拓展了三维相关研究，核心依赖SO(n-1,1)周期轨道的有效密度定理，运用了Margulis函数等多项技术。

From geometry to arithmetic of compact hyperbolic Coxeter polytopes

We establish some geometric constraints on compact Coxeter polytopes in hyperbolic spaces and show that these constraints can be a very useful tool for the classification problem of reflective anisotropic Lorentzian lattices and cocompact arithmetic hyperbolic reflection groups.

关于双曲粘合的拟算术性

On quasi-arithmeticity of hyperbolic gluings

研究广义双曲 orbifold 粘合的拟算术性，证明拟算术粘合要求构建块也拟算术且共享环境群与迹域，并据此排除了一族理想直角反棱柱的拟算术性。

Finiteness properties of automorphism spaces of manifolds with finite fundamental group

Given a closed smooth manifold $M$ of even dimension $2n\ge6$ with finite fundamental group, we show that the classifying space ${\rm BDiff}(M)$ of the diffeomorphism group of $M$ is of finite type and has finitely generated homotopy groups in every degree. We also prove a variant of this result for manifolds with boundary and deduce that the space of smooth embeddings of a compact submanifold $N\subset M$ of arbitrary codimension into $M$ has finitely generated higher homotopy groups based at the inclusion, provided the fundamental group of the complement is finite. As an intermediate result,…

Nondivergence on homogeneous spaces and rigid totally geodesics

Let $G/Γ$ be the quotient of a semisimple Lie group by an arithmetic lattice. We show that for reductive subgroups $H$ of $G$ that is large enough, the orbits of $H$ on $G/Γ$ intersect nontrivially with a fixed compact set. As a consequence, we deduce finiteness result for totally geodesic submanifolds of arithmetic quotients of symmetric spaces that do not admit nontrivial deformation and with bounded volume. Our work generalizes previous work of Tomanov--Weiss and Oh on this topic.

某些高维流形的自同构群的剩余有限性

Residual finiteness of some automorphism groups of high dimensional manifolds

本文证明了维数 $d \geq 6$ 的2-连通闭流形的拓扑映射类群是剩余有限的，并利用嵌入微积分及Weiss纤维序列等工具推导出其为算术群。

Cheeger constants of hyperbolic reflection groups and Maass cusp forms of small eigenvalues

We compute the Cheeger constants of a collection of hyperbolic surfaces corresponding to maximal non-compact arithmetic Fuchsian groups, and to subgroups which are the rotation subgroup of maximal reflection groups. The Cheeger constants are geometric quantities, but relate to the smallest eigenvalues of Maass cusp forms. From geometrical considerations, we find evidence for the existence of small eigenvalues. We search for these small eigenvalues and compute the corresponding Maass cusp forms numerically.