Embeddings of Kleinian Groups with Torsion

Rips conjectured that a non-clementary word hyperbolic group is cohopfian if and only if it is freely indecomposable. The results and examples in this paper show that cohopficity phenomenon in the case of word hyperbolic group with torsion is much more complicated than the conjecture. In particular, the cohopficity of such groups is not determined by the numbers of their ends and the cohopficity is not preserved by finite index subgroups. Our results and examples arise from Kleinian groups. Orbifold structures and orbifold maps are the new tools in our discussions. Received September, 16, 1999, Accepted September 14, 2000     

Semicharacters of Groups

We define the notion of a semicharacter of a group G: A function from the group to ?*, whose restriction to any abelian subgroup is a homomorphism. We conjecture that for any finite group, the order of the group of semicharacters is divisible by the order of the group. We prove that the conjecture holds for some important families of groups, including the Symmetric groups and the groups GL(2, q).     

On the Distribution of Galois Groups

We propose a conjecture on the distribution of number fields with given Galois group and bounded norm of the discriminant. This conjecture is known to hold for abelian groups. We give some evidence relating the general case to the composition formula for discriminants, give a heuristic argument in favor of the conjecture, and present some computational data.     

On Some Automorphisms of Orthogonal Groups in Odd Characteristic

In this paper, it is proved that the simple orthogonal groups O _2n+1(q) and O _2n ^± (q) (where q is odd) cannot be automorphism groups of finite left distributive quasigroups. This is a particular case of the conjecture stating that the automorphism group of a left distributive quasigroup is solvable. To complete the proof of the conjecture, one must test all finite groups.     

4p阶二面体群的弱3-CI性

决定了4p(p是奇素数)阶二面体群的连通3度Cayley图的完全分类,并证明4p阶二面体群不是弱3-CI群,从而否定了C.H.Li关于"所有有限群都是弱3-CI群"的猜想     

Stabilizers of Vertices of Graphs with Primitive Automorphism Groups and a Strong Version of the Sims Conjecture. III

This is the third in a series of papers whose results imply the validity of a strong version of the Sims conjecture on finite primitive permutation groups. In this paper, the case of primitive groups with simple socle of classical nonorthogonal Lie type and nonparabolic point stabilizer is considered.     

The Number of Words of a Given Length in the Planar Crystallographic Groups

This paper reveals an interrelation between the words in groups and the growth process of the corresponding graphs, based on which exact formulas for the number of words of a given length for all 17 planar crystallographic groups are obtained. Sloane’s conjecture on the graphs of these groups is proved. Bibliography: 7 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 302, 2003, pp. 188–197.     

Cores of Hyperbolic 3-Manifolds and Limits of Kleinian Groups II

Troels Jørgensen conjectured that the algebraic and geometriclimits of an algebraically convergent sequence of isomorphicKleinian groups agree if there are no new parabolics in thealgebraic limit. We prove that this conjecture holds in ‘most’cases. In particular, we show that it holds when the domainof discontinuity of the algebraic limit of such a sequence isnon-empty (see Theorem 3.1). We further show, with the sameassumptions, that the limit sets of the groups in the sequenceconverge to the limit set of the algebraic limit. As a corollary,we verify the conjecture for finitely generated Kleinian groupswhich are not (non-trivial) free products of surface groupsand infinite cyclic groups (see Corollary 3.3). These resultsare extensions of similar results for purely loxodromic groupswhich can be found in [4]. Thurston [32] previously establishedthese results in the case when the Kleinian groups are freelyindecomposable (see also Ohshika [24, 25, 27]). Using differenttechniques from ours, Ohshika [26] has proven versions of theseresults for purely loxodromic function groups.     

具有两个p'维非线性不可约特征标的非可解群

McKay猜想是有限群表示理论中的一个重要问题.本文考虑了具有两个p'维不可约特征标的非可解群群G,并证明了McKay猜想对此类群成立.更进一步,当p为奇素数时,下面结果成立:p=3,G≌PSL_2(7)或存在一个正规2-群N,满足G/N≌PSL_2(5).     

Nilpotent Blocks of Quasisimple Groups for the Prime Two

We examine nilpotency amongst blocks of positive defect of the quasisimple groups for the prime 2. We show that every nilpotent block of a quasisimple group has abelian defect groups, and prove a conjecture of Puig concerning the recognition of nilpotent blocks in the case of quasisimple groups. Explicit characterisations of nilpotent blocks are given for the classical, alternating and sporadic simple groups.     

The Novikov Conjecture for Linear Groups

Let K be a field. We show that every countable subgroup of GL(n,K) is uniformly embeddable in a Hilbert space. This implies that Novikov’s higher signature conjecture holds for these groups. We also show that every countable subgroup of GL(2,K) admits a proper, affine isometric action on a Hilbert space. This implies that the Baum-Connes conjecture holds for these groups. Finally, we show that every subgroup of GL(n,K) is exact, in the sense of C^*-algebra theory.     

Overgroups of Cyclic Sylow Subgroups of Linear Groups

We use a theorem of Guralnick, Penttila, Praeger, and Saxl to classify the subgroups of the general linear group (of a finite dimensional vector space over a finite field) which are overgroups of a cyclic Sylow subgroup. In particular, our results provide the starting point for the classification of transitive m-systems; which include the transitive ovoids and spreads of finite polar spaces. We also use our results to prove a conjecture of Cameron and Liebler on irreducible collineation groups having equally many orbits on points and on lines.     

Modular Representation Theory of Blocks with Trivial Intersection Defect Groups

We show that Uno's refinement of the projective conjecture of Dade holds for every block whose defect groups intersect trivially modulo the maximal normal p-subgroup. This corresponds to the block having p-local rank one as defined by Jianbei An and Eaton. An immediate consequence is that Dade's projective conjecture, Robinson's conjecture, Alperin's weight conjecture, the Isaacs–Navarro conjecture, the Alperin–McKay conjecture and Puig's nilpotent block conjecture hold for all trivial intersection blocks. Presented by A. Verschoren Mathematics Subject Classification (2000) Primary 20C20. Charles W. Eaton: Current address: School of Mathematics, University of Manchester, Sackville Street, PO Box 88, Manchester M60 1QC, U.K. e-mail: charles.eaton@manchester.ac.uk This research was supported in part by the Marsden Fund of New Zealand via grant #9144/3368248.     

Characteristics of Graph Braid Groups

We give formulae for the first homology of the n-braid group and the pure 2-braid group over a finite graph in terms of graph-theoretic invariants. As immediate consequences, a graph is planar if and only if the first homology of the n-braid group over the graph is torsion-free and the conjectures about the first homology of the pure 2-braid groups over graphs in Farber and Hanbury (arXiv:1005.2300 [math.AT]) can be verified. We discover more characteristics of graph braid groups: the n-braid group over a planar graph and the pure 2-braid group over any graph have a presentation whose relators are words of commutators, and the 2-braid group and the pure 2-braid group over a planar graph have a presentation whose relators are commutators. The latter was a conjecture in Farley and Sabalka (J. Pure Appl. Algebra, 2012) and so we propose a similar conjecture for higher braid indices.     

Free Subgroups of Groups with Nontrivial Floyd Boundary

Abstract

We prove that when a countable group admits a nontrivial Floyd-type boundary, then every nonelementary and metrically proper subgroup contains a noncommutative free subgroup. This generalizes the corresponding well-known results for hyperbolic groups and groups with infinitely many ends. It also shows that no finitely generated amenable group admits a nontrivial boundary of this type. This improves on a theorem by Floyd (Floyd, W. J. (1980). Group completions and limit sets of Kleinian groups. Invent. Math. 57: 205–218) as well as giving an elementary proof of a conjecture stated in that same paper. We also show that if the Floyd boundary of a finitely generated group is nontrivial, then it is a boundary in the sense of Furstenberg and the group acts on it as a convergence group.     

The Alperin and Dade Conjectures for the O'Nan and Rudivalis Simple Groups

We classify the radical subgroups and chains of the O'Nan and Rudvalis simple groups O'N and Ru, and then verify the Alperin weight conjecture and the Dade final conjecture for the two groups.     

An Obstruction to Fundamental Groups of Positively Ricci Curved Manifolds

In this note we propose a conjecture concerning fundamental groups of Riemannian n-manifolds with positive Ricci curvature. We prove a partial result under an extra condition on a lower bound of sectional curvature. Our main tool is the theory of Hausdorff convergence. We also extend Fukaya and Yamaguchi's resolution of a conjecture of Gromov to limit spaces which may have singular points.     

A New Approach to Extracting Roots in Garside Groups

We study the problem of extracting roots in Garside groups by reducing it to the calculation of certain ultra summit sets. Several properties concerning the roots and an effective algorithm are derived. In particular, in the case of braid groups, a conjecture on the bound of super summit set implies that, for fixed number of strands and ordinal number of root, the algorithm is polynomial in the word length.     

Recognizing Finite Groups Through Order and Degree Patterns

The degree pattern of a finite group G associated with its prime graph has been introduced by Moghaddamfar in 2005 and it is proved that the following simple groups are uniquely determined by their order and degree patterns:All sporadic simple groups,the alternating groups Ap(p≥5 is a twin prime)and some simple groups of the Lie type.In this paper,the authors continue this investigation.In particular,the authors show that the symmetric groups Sp+3,where p+2 is a composite number and p+4 is a prime and 97相似文献