Some arithmetic groups that do not act on the circle

In this final chapter we extend our knowledge of group theory. Among other aspects of finite groups, we investigate permutation groups and obtain two results of cardinal importance in the theory of finite groups. In the first of these, we establish the structure of Abelian groups and, in the second, we establish the existence of the so-called ‘Sylow p-subgroups’ of a finite group.KeywordsAbelian GroupNormal SubgroupFinite GroupCyclic GroupSymmetric GroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

CONTENTS Introduction § 1. Arithmetic groups § 2. Adèle groups § 3. Tamagawa numbers § 4. Approximations in algebraic groups § 5. Class numbers and class groups of algebraic groups § 6. The genus problem in arithmetic groups § 7. Classification of maximal arithmetic subgroups § 8. The congruence problem § 9. Groups of rational points over global fields § 10. Galois cohomology and the Hasse principle § 11. Cohomology of arithmetic groups References

In a previous paper, we produced a “Jordan decomposition of characters” for the groupsGL n (O), whereO is the ring of integers in ap-adic field, splitting the representation theory into “nilpotent” and “semisimple” cases. This paper is concerned with the nilpotent representations. This problem is closely related to both the representation theory of the fullp-adic group and the arithmetic group over the ring of integers of a global field. The main result is an explicit description of the irreducible representations of the endomorphism ring, or Hecke algebra, of an induced nilpotent representation. Our methods are combinatorial and along the way we study a certainp-adic generalization of theq-Johnson association scheme. Our Hecke algebra is isomorphic to its Bose-Mesner algebra. The character values of the irreducible representations of this algebra givep-adic generalizations of theq-Hahn polynomials.

We study certain polynomial trace identities in the group SL(2, C) and their application in the theory of discrete groups. We obtain canonical representations for two generator groups in §4, and then in §5 we give a new proof for Gehring and Martin's polynomial trace identities for good words and extend that result to a larger class which is also closed under a semigroup operation inducing polynomial composition. This new approach is through the use of quaternion algebras over indefinite and an associated group of units. We obtain structure theorems for these quaternion algebras, which appear to be of independent interest in §8. Using these quaternion algebras and their units, we consider their relation to discrete subgroups of SL(2, C), giving necessary and sufficient criteria for discreteness and another for arithmeticity §9. We then show that for the groups Zp ∗ Z2, the complement of the closure of roots of the good word polynomials is precisely the moduli space of geometrically finite discrete and faithful representations a result we show holds in greater generality in §12.

A dichotomy for integral group rings via higher modular groups as amalgamated products

This meeting was focused on asymptotic aspects of group theory. The resulting problems lead, in particular, to the study of infinite groups, with an emphasis on the asymptotic behaviour of their finite quotients. Also, properties of infinite families of finite groups are at the center of interest in the field. We had talks on the most important recent developments in the field of asymptotic and profinite group theory. We single out some of these new results. The classification of finite simple groups has provided an impetus for the detailed investigation of properties of families of groups of Lie type and of alternating groups. This has produced many results of asymptotic or probabilistic nature, describing in a quantitative way properties that hold for all sufficiently large simple groups, or else that hold ‘almost surely’ as the orders of the groups are tending to infinity. Here we had a talk of Aner Shalev on his recent solution (with coauthors) of Ore's conjecture. Recently there has been much activity in the study of infinite groups, with an emphasis on the asymptotic behaviour of their finite quotients, or their finite-index subgroups. The theory of subgroup growth is now well established; the theory of the associated zeta functions is making good progress. More subtle variations of subgroup growth are also developing into significant theories in their own right: there are important new results on representation growth – of arithmetic groups, nilpotent groups and pro-p groups; and on maximal subgroup growth, with applications to probabilistic generation properties. Profinite groups are ‘asymptotic limits’ of finite groups. The nature of verbal subgroups and verbal mappings, in both finite and profinite groups, has been the focus of much recent work. This has led among other things to the solution of Serre's problem on subgroups of finite index in finitely generated profinite groups; other significant results include the characterisation of closed verbal subgroups in pro-p groups and new characterisations of solubility in finite groups. Branch groups, both discrete and profinite, are certain groups of automorphisms of rooted trees; introduced by Grigorchuk and others over 20 years ago, they have provided the solution to several outstanding problems, and their study continues to provide new results. Some of them were reported on in our meeting. We also had talks from important fields of mathematics in which profinite and asymptotic group theory finds application. There is, for instance, the study of the absolute Galois group of the rationals and in particular its closed pro-p subgroups. Methods from pro-p group theory have successfully been applied in order to understand fundamental groups of (hyperbolic) 3-manifolds. The interplay between the topological fundamental group of a complex algebraic surface and its profinite completion which is the etale fundamental group of the surface is the subject of recent research. We believe that the meeting was exciting and successful. The high quality lectures covered a lot of very recent results in the field. There were many intense discussions between mathematicians of different fields. Altogether there were over 50 participants from all over the world. The percentage of young researchers was very high. The following extended abstracts were collected and edited by Dr. Evija Ribnere.

In [6], p. 80, Serre showed that the group SL2(Z[1/p]) is a free product of two copies of SL2(Z) amalgamated along a sub-group of index p + 1 in both components. This result was obtained from the structure theory of groups acting on trees and the construction of a tree on which SL2 over a field with discrete valuation acts in a natural way. This result can be extended to other SL2 examples. Using the above description, Serre [6] also gave presentations for the groups SL2(Z[1/p]) where p = 2, 3.

The XXXth International Colloquium on Group Theoretical Methods in Physics (ICGTMP), also known as the Group30 conference, took place in Ghent (Belgium) from Monday 14 to Friday 18 July 2014. The conference was organised by Ghent University (Department of Applied Mathematics, Computer Science and Statistics, and Department of Mathematical Analysis). The website http://www.group30.ugent.be is still available.

A metric tree is a metric space such that between any two points there is a unique topological segment (i.e., a subset homeomorphic to an interval of the real line), and such that this segment is a geodesic (i.e., the isometric image of an interval). The theory of group actions on metric trees and of the structure of these trees on the boundary at infinity has many different aspects, and it has been made clear that this theory captures a lot of the interesting phenomena of hyperbolic geometry, or, to be more precise, of the theory of groups acting by isometrics on the hyperbolic space I/~ and on its sphere at infinity. One of the particular features, in the case of metric trees, is that the ideas and methods which are used are often of a combinatorial nature, and, probably, simpler and more intuitive than those of classical hyperbolic geometry. In particular, in [2] and [3], we give a series of criteria for the ergodicity of the geodesic flow associated with groups acting properly and isometrically on metric trees, which are discrete analogs of theorems due to Hopf and Sullivan in the case of hyperbolic manifolds. The invariant measures on these flows come from conformal densities on the boundary. This theory of conformal densities on the boundary was developed, in the case of hyperbolic manifolds, by Patterson and Sullivan (see [5] and [6]), and later, in the case of metric trees, in [11 . The result that we prove here concerns a Fatou-type behavior of certain natural functions associated with conformal densities on the boundary of a metric tree. A somehow more special result of this kind, in the case of conformal densities on the boundary of the hyperbolic n-space, is used by Sullivan in [6], in his proof for the probabilistic criterion of the ergodicity of the geodesic flow on a hyperbolic manifold (see also [7, Theorem 2.14]). T h e p r o o f of the corresponding result for metric trees, which we give in [31, involves a different kind of argument, although the Fatou-type behavior is also valid in that case, and, indeed, the aim of the present paper is to prove this result in the case of metric trees. Let us begin by describing the background. Let X be a metric tree which is complete and locally compact. We denote the distance between two points in X by Ix YlA geodesic ray (resp. a geodesic segment) is a map f from [0, oo[ (resp. from a closed interval) into X such that I f(r t ) f(r2)l = Irt r~l for every r, and r2. The boundary OX is the space of the ends of X. Every geodesic ray r defines a point in OX, which is denoted by r(oo). Let us also recall that between any two points x E X and r /E OX, there is a unique image of a geodesic ray, which is denoted by Ix, r/[. There is a natural topology on X U OX, which can easily be defined with the use of the ray structure from any given point in X, and for this topology, the space X U OX is compact and X lies in i t as an open dense subset. Better than that, X tO OX is, in fact, equipped with a natural class of metrics, and the induced class of metrics on OX, called visual metrics, is most useful. Let us recall its definition: For every point z E X, the visual metric I I~ on OX is defined by

We show that large classes of non-arithmetic hyperbolic n -manifolds, including the hybrids introduced by Gromov and Piatetski-Shapiro and many of their generalizations, have only finitely many finite-volume immersed totally geodesic hypersurfaces. In higher codimension, we prove finiteness for geodesic submanifolds of dimension at least 2 that are maximal , i.e., not properly contained in a proper geodesic submanifold of the ambient n -manifold. The proof is a mix of structure theory for arithmetic groups, dynamics, and geometry in negative curvature.

Book reviews

Let \(B\) be a class of groups. The elementary class with base \(B\) is defined as the minimal class of groups containing \(B\) and closed with respect to taking subgroups, quotient groups, group extensions, and direct limits. Properties of such classes are studied. Some applications to the theory of elementary amenable groups and a relation to the Kurosh--Chernikov classes of generalized solvable groups are considered.

These are the notes of a five-lecture course presented at the Computations with Modular Forms summer school, aimed at graduate students in number theory and related areas. Sections 1–4 give a sketch of the theory of reductive algebraic groups over Q, and of Gross’s purely algebraic definition of automorphic forms in the special case when G(R) is compact. Sections 5–9 describe how these automorphic forms can be explicitly computed, concentrating on the case of definite unitary groups; and Sects. 10 and 11 describe how to relate the results of these computations to Galois representations, and present some examples where the corresponding Galois representations can be identified, giving illustrations of various instances of Langlands’ functoriality conjectures.KeywordsModular FormAlgebraic GroupReductive GroupCompact SubgroupAutomorphic FormThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The cohomology of arithmetic groups is made up of two pieces, the cuspidal and noncuspidal parts. Within the cuspidal cohomology is a subspace— the f -cuspidal cohomology—spanned by the classes that generate representations of the associated finite Lie group which are cuspidal in the sense of finite Lie group theory. Few concrete examples of f -cuspidal cohomology have been computed geometrically, outside the cases of rational rank 1, or where the symmetric space has a Hermitian structure. This paper presents new computations of the f -cuspidal cohomology of principal congruence subgroups Γ ( p ) \Gamma (p) of GL ( 3 , Z ) {\text {GL}}(3,\mathbb {Z}) of prime level p . We show that the f -cuspidal cohomology of Γ ( p ) \Gamma (p) vanishes for all p ⩽ 19 p \leqslant 19 with p ≠ 11 p \ne 11 , but that it is nonzero for p = 11 p = 11 . We give a precise description of the f -cuspidal cohomology for Γ ( 11 ) \Gamma (11) in terms of the f -cuspidal representations of the finite Lie group GL ( 3 , Z / 11 ) {\text {GL}}(3,\mathbb {Z}/11) . We obtained the result, ultimately, by proving that a certain large complex matrix M is rank-deficient. Computation with the SVD algorithm gave strong evidence that M was rank-deficient; but to prove it, we mixed ideas from numerical analysis with exact computation in algebraic number fields and finite fields.

Although in 1919 I was an undergraduate at the University of Wisconsin, my parents properly felt that it would be a salutary experience for me to spend a summer on another campus. At that time, the University of Chicago had the preeminent graduate summer school in physics; so I went there at the end of my junior year, and took as part of my program a course in optics under Professor Michelson. Although he was 67 years old, he was vigorous both mentally and physically, even playing tennis at the Quadrangle Club. He had a pedagogical system of his own. There was no final written examination; instead, every week he would interrupt his lectures to send some quivering student to the blackboard to explain a topic that he, Michelson, had previously presented. This system has much to be said in its favor. It insures that the students keep abreast of the lecture material and are clear in their exposition. I can remember that when Michelson felt the presentation was not lucid, he made some such remark as, “How would you expect a beginner to understand what you are saying?”