Kai-Uwe Bux
We give an overview about finiteness properties of soluble S-arithmetic groups. Both, the number field case and the function field case are covered. The main result is: If B is a Borel subgroup in a Chevalley group and R is an S-arithmetic ring, then the group B(R) has finiteness length |S|-1 in the function field case, and infinite finiteness length in the number field case.
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We prove that there are only finitely many conjugacy classes of arithmetic maximal hyperbolic reflection groups.
We determine when an arithmetic subgroup of a reductive group defined over a global function field is of type FP_\infty by comparing its large-scale geometry to the large-scale geometry of lattices in real semisimple Lie groups.
We show that the finiteness length of an $S$-arithmetic subgroup $Γ$ in a noncommutative isotropic absolutely almost simple group $G$ over a global function field is one less than the sum of the local ranks of $G$ taken over the places in $S$. This determines the finiteness properties for arithmetic subgroups in isotropic reductive groups, confirming the conjectured finiteness properties for this class of groups. Our main tool is Behr-Harder reduction theory which we recast in terms of the metric structure of euclidean buildings.
Let G be a Chevalley group scheme and B<=G a Borel subgroup scheme, both defined over Z. Let K be a global function field, S be a finite non-empty set of places over K, and O_S be the corresponding S-arithmetic ring. Then, the S-arithmetic group B(O_S) is of type F_{|S|-1} but not of type FP_{|S|}. Moreover one can derive lower and upper bounds for the geometric invariants Σ^m(B(O_S)). These are sharp if G has rank 1. For higher ranks, the estimates imply that normal subgroups of B(O_S) with abelian quotients, generically, satisfy strong finiteness conditions.
Finiteness of closed arithmetic hyperbolic surface bundles
本文证明了对每个亏格 g≥2，闭纤维化算术双曲 3-流形仅有有限多个循环可公度类，并给出有效上界，证实了 Bowditch-Maclachlan-Reid 猜想。
This paper is a follow-up to our joint paper with I. Agol, P. Storm and K. Whyte "Finiteness of arithmetic hyperbolic reflection groups". 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 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.
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…
The normal Farb growth of a group quantifies how well-approximated the group is by its finite quotients. We show that any S-arithmetic subgroup of a higher rank Chevalley group G has normal Farb growth n^dim(G).
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…
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…
Let $Γ$ be an arithmetic subgroup of $SU(d,1)$ with cusps, and let $X_Γ$ be the associated locally symmetric space. We prove that if the first inner cohomology group $H^1_!(X_Γ,\mathbb{C})$ is non-zero then the pre-image of $Γ$ in each connected cover of $SU(d,1)$ is residually finite. We also give an example of a such a group $Γ$ for which $H^1_!(X_Γ,\mathbb{C})$ is non-zero.
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…
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.
We provide partial results towards a conjectural generalization of a theorem of Lubotzky-Mozes-Raghunathan for arithmetic groups (over number fields or function fields) that implies, in low dimensions, both polynomial isoperimetric inequalities and finiteness properties. As a tool in our proof, we establish polynomial isoperimetric inequalities and finiteness properties for certain solvable groups that appear as subgroups of parabolic groups in semisimple groups, thus generalizing a theorem of Bux. We also develop a precise version of reduction theory for arithmetic groups whose proof is, for …
We show that the homological finiteness length of a non-uniform lattice on a locally finite CAT(0) n-dimensional polyhedral complex is less than n. As a corollary, we obtain an upper bound for the homological finiteness length of arithmetic groups over function fields. This gives an easier proof of a result of Bux and Wortman that solved a long-standing conjecture.
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。
Residual finiteness and cuspidal cohomology of Picard modular surfaces
本文证明SU(2,1)中非均匀算术格在万有覆盖及连通有限覆盖中的逆像均剩余有限，关键新输入是每个可公度类含同余格使尖点第一上同调非零，证明基于Rogawski内窥镜分类及作者先前上同调判据。
We introduce the concept of quantifying the extent to which a finitely generated group is residually finite. The quantification is carried out for some examples including free groups, the first Grigorchuk group, finitely generated nilpotent groups, and certain arithmetic groups such as $SL_n(\mathbb{Z})$. In the context of finite nilpotent quotients, we find a new characterization of nilpotent groups.
This is a survey article on some recent developments in the arithmetic theory of linear algebraic groups over higher-dimensional fields, written for the Notices of the AMS.
Ascending chains of irreducible lattices, bi-reversible automata and affine arithmetic groups
研究在\(n\)个齐次树乘积中构造不可约格升链，以及定义双可逆自动机使其定义的群有特定有限性长度，核心方法是借助全局函数域仿射群中\(S -\)算术群，贡献是实现相关构造。
We give a criterion for bounding the homological finiteness length of certain HF-groups. This is used in two distinct contexts. Firstly, the homological finiteness length of a non-uniform lattice on a locally finite n-dimensional contractible CW-complex is less than n. In dimension two it solves a conjecture of Farb, Hruska and Thomas. As another corollary, we obtain an upper bound for the homological finiteness length of arithmetic groups over function fields. This gives an easier proof of a result of Bux and Wortman that solved a long-standing conjecture. Secondly, the criterion is applied t…
Finiteness properties of algebraic fibers of group extensions
本综述总结了群扩张的代数纤维化存在性及其代数纤维的有限性性质，并探讨了这些结果在抽象群、pro-p群及高阶相干性中的应用。
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。
Residual finiteness growth gives an invariant that indicates how well-approximated a finitely generated group is by its finite quotients. We briefly survey the state of the subject. We then improve on the best known upper and lower bounds for lamplighter groups. Notably, any lamplighter group has super-linear residual finiteness growth. In our proof, we quantify a congruence subgroup property for lamplighter groups.
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。
Automorphisms of Bestvina-Brady Groups: IA Rigidity, Arithmetic Commensurability, and Finiteness
本文研究了Bestvina-Brady群的自同构性质，证明了IA刚性、算术共轭性和有限性相关结论，并探讨了其外自同构群的有限性。
On the finiteness length of some soluble linear groups
本文研究含单位元交换环上可解线性群的有限性长度，通过亚阿贝尔像的R-表示给出群有限性长度的上界，结合已有结果给出Bux等式推广的新证明，还给出Strebel未发表定理的替代证明并推广早期结果、引出相关猜想。
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.
We survey general properties of multiplicative arithmetic functions of several variables and related convolutions, including the Dirichlet convolution and the unitary convolution. We introduce and investigate a new convolution, called gcd convolution. We define and study the convolutes of arithmetic functions of several variables, according to the different types of convolutions. We discuss the multiple Dirichlet series and Bell series and present certain arithmetic and asymptotic results of some special multiplicative functions arising from problems in number theory, group theory and combinat…
Let $G$ be a finite group and $Ch_i(G)$ some quantitative sets. In this paper we study the influence of $Ch_i(G)$ to the structure of $G$. We present a survey of author and his colleagues' recent works.
Residual Finiteness Growth in Virtually Abelian Groups
本文研究几乎阿贝尔群的剩余有限性增长，证明其等于 $\log^k$，其中 $k$ 由显式表达式给出，并应用于构造具有指定增长阶的群。
We generalize a result of Sury and prove that uniform discreteness of cocompact lattices in higher rank semisimple Lie groups (first conjectured by Margulis) is equivalent to a weak form of Lehmer's conjecture. We include a short survey of related results and conjectures.
This paper makes a comprehensive survey of results relating to finite Rees index for semigroups. In particular, we survey of the state of knowledge on whether various finiteness properties (such as finite generation, finite presentability, automaticity, and hopficity) are inherited by finite Rees index subsemigroups and extensions. We survey other properties that are invariant under passing to finite Rees index subsemigroups and extensions, such as the cofinality and number of ends. We prove some new results: inheritance of word-hyperbolicity by finite Rees index subsemigroups, and inheritance…
We present a survey of results on word equations in simple groups, as well as their analogues and generalizations, which were obtained over the past decade using various methods, group-theoretic and coming from algebraic and arithmetic geometry, number theory, dynamical systems and computer algebra. Our focus is on interrelations of these machineries which led to numerous spectacular achievements, including solutions of several long-standing problems.
We discuss several arithmetic aspects of Bianchi groups, especially from a computational point of view. In particular, we consider computing the homology of Bianchi groups together with the Hecke action, connections with automorphic forms, abelian varieties, Galois representations and the torsion in the homology of Bianchi groups. Along the way, we list several open problems and conjectures, survey the related literature, presenting concrete examples and numerical data.
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