Analogues of Cayley graphs for topological groups

We define for a compactly generated totally disconnected locally compact group a graph, called a rough Cayley graph, that is a quasi-isometry invariant of the group. This graph carries information about the group structure in an analogous way to the ordinary Cayley graph for a finitely generated group. With this construction the machinery of geometric group theory can be applied to topological groups. This is illustrated by a study of groups where the rough Cayley graph has more than one end and a study of groups where the rough Cayley graph has polynomial growth. Supported by project J2245 of the Austrian Science Fund (FWF) and be an IEF Marie Curie Fellowship of the Commission of the European Union.     

The non-orientable genus of some metacyclic groups

We describe non-orientable, octagonal embeddings for certain 4-valent, bipartite Cayley graphs of finite metacyclic groups, and give a class of examples for which this embedding realizes the non-orientable genus of the group. This yields a construction of Cayley graphs for which is arbitrarily large, where and are the orientable genus and the non-orientable genus of the Cayley graph.Work supported in part by the Research Council of Slovenia, Yugoslavia and NSF Contract DMS-8717441.Supported by NSF Contract DMS-8601760.     

On the structure of surface pure braid groups

We describe some of the properties of the pure braid groups of surfaces different from and . In the case of compact, connected, orientable surfaces without boundary and of genus at least two, we give a necessary and sufficient condition for the splitting of the pure braid group exact sequence of Fadell and Neuwirth, thus answering completely a question of Birman.     

On the structure of surface pure braid groups

We describe some of the properties of the pure braid groups of surfaces different from and . In the case of compact, connected, orientable surfaces without boundary and of genus at least two, we give a necessary and sufficient condition for the splitting of the pure braid group exact sequence of Fadell and Neuwirth, thus answering completely a question of Birman.     

Subword complexes in Coxeter groups

Let (Π,Σ) be a Coxeter system. An ordered list of elements in Σ and an element in Π determine a subword complex, as introduced in Knutson and Miller (Ann. of Math. (2) (2003), to appear). Subword complexes are demonstrated here to be homeomorphic to balls or spheres, and their Hilbert series are shown to reflect combinatorial properties of reduced expressions in Coxeter groups. Two formulae for double Grothendieck polynomials, one of which appeared in Fomin and Kirillov (Proceedings of the Sixth Conference in Formal Power Series and Algebraic Combinatorics, DIMACS, 1994, pp. 183-190), are recovered in the context of simplicial topology for subword complexes. Some open questions related to subword complexes are presented.     

A new approach to the construction of subsystems of complex root systems

Dynkin has shown how subsystems of real root systems may be constructed. As the concept of subsystems of complex root systems is not as well developed as in the real case, in this paper we give an algorithm to classify the proper subsystems of complex proper root systems. Furthermore, as an application of this algorithm, we determine the proper subsystems of imprimitive complex proper root systems. These proper subsystems are useful in giving combinatorial constructions of irreducible representations of properly generated finite complex reflection groups.     

Robinson-Schensted algorithm and Vogan equivalence

We provide a combinatorial proof for the coincidence of Knuth equivalence classes, Kazhdan-Lusztig left cells and Vogan classes for the symmetric group, involving only Robinson-Schensted algorithm and the combinatorial part of the Kazhdan-Lusztig cell theory.     

Coxeter groups are virtually special

In this paper we prove that every finitely generated Coxeter group has a finite index subgroup that is the fundamental group of a special cube complex. Some consequences include: Every f.g. Coxeter group is virtually a subgroup of a right-angled Coxeter group. Every word-hyperbolic Coxeter group has separable quasiconvex subgroups.     

Twisted conjugacy classes in nilpotent groups

Let N be a finitely generated nilpotent group. We show that there is an algorithm that for any automorphism φ∈Aut(N) computes its Reidemeister number R(φ). It is proved that any free nilpotent group N_rc of rank r and class c belongs to class R_∞ if any of the following conditions holds: r=2 and c≥4; r=3 and c≥12; r≥4 and c≥2r.     

Automorphisms of groups

The survey presents classical assertions due to Nielsen, Whitehead, and others, well-known theorems on automorphisms included in monographs on group theory, and recent results in this area. Attention is focused on the progress in automorphism groups theory for free, solvable, modular, and profinite groups. New tools of investigation using graphs and geometrical ideas are also discussed.     

The axiomatic closure of the class of discriminating groups

In [6] squarelike groups were defined to be those groups G universally equivalent to their direct squares G × G. In that paper it was shown that G is squarelike if and only if G is universally equivalent to a discriminating group in the sense of [3]. Further it was shown that the class of squarelike groups is first-order axiomatizable while the class of discriminating groups is not. In this paper, we prove that the class of squarelike groups is the least axiomatic class containing the discriminating groups.Received: 18 August 2003     

Weights of multipartitions and representations of Ariki-Koike algebras

We consider representations of the Ariki-Koike algebra, a q-deformation of the group algebra of the complex reflection group C_r?S_n. The representations of this algebra are naturally indexed by multipartitions of n, and for each multipartition λ we define a non-negative integer called the weight of λ. We prove some basic properties of this weight function, and examine blocks of small weight.     

Subgroup distortion in wreath products of cyclic groups

We study the effects of subgroup distortion in the wreath products , where A is finitely generated abelian. We show that every finitely generated subgroup of has distortion function equivalent to some polynomial. Moreover, for A infinite, and for any polynomial l^k, there is a 2-generated subgroup of having distortion function equivalent to the given polynomial. Also, a formula for the length of elements in arbitrary wreath product easily shows that the group has distorted subgroups, while the lamplighter group has no distorted (finitely generated) subgroups. In the course of the proof, we introduce a notion of distortion for polynomials. We are able to compute the distortion of any polynomial in one variable over Z,R or C.     

On infinite tournaments with regular automorphism groups

Atournament regular representation (TRR) of an abstract groupG is a tournamentT whose automorphism group is isomorphic toG and is a regular permutation group on the vertices ofT. L. Babai and W. Imrich have shown that every finite group of odd order exceptZ ₃ ×Z ₃ admits a TRR. In the present paper we give several sufficient conditions for an infinite groupG with no element of order 2 to admit a TRR. Among these are the following: (1)G is a cyclic extension byZ of a finitely generated group; (2)G is a cyclic extension byZ _2n+1 of any group admitting a TRR; (3)G is a finitely generated abelian group; (4)G is a countably generated abelian group whose torsion subgroup is finite.     

Rouquier blocks

This paper investigates the Rouquier blocks of the Hecke algebras of the symmetric groups and the Rouquier blocks of the q-Schur algebras. We first give an algorithm for computing the decomposition numbers of these blocks in the ``abelian defect group case' and then use this algorithm to explicitly compute the decomposition numbers in a Rouquier block. For fields of characteristic zero, or when q=1 these results are known; significantly, our results also hold for fields of positive characteristic with q≠1. We also discuss the Rouquier blocks in the ``non–abelian defect group' case. Finally, we apply these results to show that certain Specht modules are irreducible.     

Minimal Bar Tableaux

Motivated by recent results of Stanley, we generalize the rank of a partition λ to the rank of a shifted partition S(λ). We show that the number of bars required in a minimal bar tableau of S(λ) is max(o, e + (ℓ(λ) mod 2)), where o and e are the number of odd and even rows of λ. As a consequence we show that the irreducible projective characters of S_n vanish on certain conjugacy classes. Another corollary is a lower bound on the degree of the terms in the expansion of Schur’s Q_λ symmetric functions in terms of the power sum symmetric functions. Received November 20, 2003