51. CJM 2003 (vol 55 pp. 1080)
 Kellerhals, Ruth

Quaternions and Some Global Properties of Hyperbolic $5$Manifolds
We provide an explicit thick and thin decomposition for oriented
hyperbolic manifolds $M$ of dimension $5$. The result implies improved
universal lower bounds for the volume $\rmvol_5(M)$ and, for $M$
compact, new estimates relating the injectivity radius and the diameter
of $M$ with $\rmvol_5(M)$. The quantification of the thin part is
based upon the identification of the isometry group of the universal
space by the matrix group $\PS_\Delta {\rm L} (2,\mathbb{H})$ of
quaternionic $2\times 2$matrices with Dieudonn\'e determinant
$\Delta$ equal to $1$ and isolation properties of $\PS_\Delta {\rm
L} (2,\mathbb{H})$.
Categories:53C22, 53C25, 57N16, 57S30, 51N30, 20G20, 22E40 

52. CJM 2003 (vol 55 pp. 750)
 Göbel, Rüdiger; Shelah, Saharon; Strüngmann, Lutz

AlmostFree $E$Rings of Cardinality $\aleph_1$
An $E$ring is a unital ring $R$ such that every endomorphism of
the underlying abelian group $R^+$ is multiplication by some
ring element. The existence of almostfree $E$rings of
cardinality greater than $2^{\aleph_0}$ is undecidable in $\ZFC$.
While they exist in G\"odel's universe, they do not exist in other
models of set theory. For a regular cardinal $\aleph_1 \leq
\lambda \leq 2^{\aleph_0}$ we construct $E$rings of cardinality
$\lambda$ in $\ZFC$ which have $\aleph_1$free additive structure.
For $\lambda=\aleph_1$ we therefore obtain the existence of
almostfree $E$rings of cardinality $\aleph_1$ in $\ZFC$.
Keywords:$E$rings, almostfree modules Categories:20K20, 20K30, 13B10, 13B25 

53. CJM 2003 (vol 55 pp. 822)
 Kim, Djun Maximilian; Rolfsen, Dale

An Ordering for Groups of Pure Braids and FibreType Hyperplane Arrangements
We define a total ordering of the pure braid groups which is
invariant under multiplication on both sides. This ordering is
natural in several respects. Moreover, it wellorders the pure braids
which are positive in the sense of Garside. The ordering is defined
using a combination of Artin's combing technique and the Magnus
expansion of free groups, and is explicit and algorithmic.
By contrast, the full braid groups (on 3 or more strings) can be
ordered in such a way as to be invariant on one side or the other, but
not both simultaneously. Finally, we remark that the same type of
ordering can be applied to the fundamental groups of certain complex
hyperplane arrangements, a direct generalization of the pure braid
groups.
Category:20F36 

54. CJM 2002 (vol 54 pp. 1229)
55. CJM 2002 (vol 54 pp. 970)
56. CJM 2002 (vol 54 pp. 795)
 Möller, Rögnvaldur G.

Structure Theory of Totally Disconnected Locally Compact Groups via Graphs and Permutations
Willis's structure theory of totally disconnected locally compact groups
is investigated in the context of permutation actions. This leads to new
interpretations of the basic concepts in the theory and also to new proofs
of the fundamental theorems and to several new results. The treatment of
Willis's theory is selfcontained and full proofs are given of all the
fundamental results.
Keywords:totally disconnected locally compact groups, scale function, permutation groups, groups acting on graphs Categories:22D05, 20B07, 20B27, 05C25 

57. CJM 2001 (vol 53 pp. 1121)
 Athanasiadis, Christos A.; Santos, Francisco

Monotone Paths on Zonotopes and Oriented Matroids
Monotone paths on zonotopes and the natural generalization to maximal
chains in the poset of topes of an oriented matroid or arrangement of
pseudohyperplanes are studied with respect to a kind of local move,
called polygon move or flip. It is proved that any monotone path on a
$d$dimensional zonotope with $n$ generators admits at least $\lceil
2n/(nd+2) \rceil1$ flips for all $n \ge d+2 \ge 4$ and that for any
fixed value of $nd$, this lower bound is sharp for infinitely many
values of $n$. In particular, monotone paths on zonotopes which admit
only three flips are constructed in each dimension $d \ge 3$.
Furthermore, the previously known 2connectivity of the graph of
monotone paths on a polytope is extended to the 2connectivity of the
graph of maximal chains of topes of an oriented matroid. An
application in the context of Coxeter groups of a result known to be
valid for monotone paths on simple zonotopes is included.
Categories:52C35, 52B12, 52C40, 20F55 

58. CJM 2000 (vol 52 pp. 1310)
 Yagunov, Serge

On the Homology of $\GL_n$ and Higher PreBloch Groups
For every integer $n>1$ and infinite field $F$ we construct a spectral
sequence converging to the homology of $\GL_n(F)$ relative to the
group of monomial matrices $\GM_n(F)$. Some entries in $E^2$terms of
these spectral sequences may be interpreted as a natural
generalization of the Bloch group to higher dimensions. These groups
may be characterized as homology of $\GL_n$ relatively to $\GL_{n1}$
and $\GM_n$. We apply the machinery developed to the investigation of
stabilization maps in homology of General Linear Groups.
Categories:19D55, 20J06, 18G60 

59. CJM 2000 (vol 52 pp. 1018)
 Reichstein, Zinovy; Youssin, Boris

Essential Dimensions of Algebraic Groups and a Resolution Theorem for $G$Varieties
Let $G$ be an algebraic group and let $X$ be a generically free $G$variety.
We show that $X$ can be transformed, by a sequence of blowups with smooth
$G$equivariant centers, into a $G$variety $X'$ with the following
property the stabilizer of every point of $X'$ is isomorphic to a
semidirect product $U \sdp A$ of a unipotent group $U$ and a
diagonalizable group $A$.
As an application of this result, we prove new lower bounds on essential
dimensions of some algebraic groups. We also show that certain
polynomials in one variable cannot be simplified by a Tschirnhaus
transformation.
Categories:14L30, 14E15, 14E05, 12E05, 20G10 

60. CJM 2000 (vol 52 pp. 449)
61. CJM 2000 (vol 52 pp. 265)
 Brion, Michel; Helminck, Aloysius G.

On Orbit Closures of Symmetric Subgroups in Flag Varieties
We study $K$orbits in $G/P$ where $G$ is a complex connected
reductive group, $P \subseteq G$ is a parabolic subgroup, and $K
\subseteq G$ is the fixed point subgroup of an involutive
automorphism $\theta$. Generalizing work of Springer, we
parametrize the (finite) orbit set $K \setminus G \slash P$ and we
determine the isotropy groups. As a consequence, we describe the
closed (resp. affine) orbits in terms of $\theta$stable
(resp. $\theta$split) parabolic subgroups. We also describe the
decomposition of any $(K,P)$double coset in $G$ into
$(K,B)$double cosets, where $B \subseteq P$ is a Borel subgroup.
Finally, for certain $K$orbit closures $X \subseteq G/B$, and for
any homogeneous line bundle $\mathcal{L}$ on $G/B$ having nonzero
global sections, we show that the restriction map $\res_X \colon
H^0 (G/B, \mathcal{L}) \to H^0 (X, \mathcal{L})$ is surjective and
that $H^i (X, \mathcal{L}) = 0$ for $i \geq 1$. Moreover, we
describe the $K$module $H^0 (X, \mathcal{L})$. This gives
information on the restriction to $K$ of the simple $G$module $H^0
(G/B, \mathcal{L})$. Our construction is a geometric analogue of
Vogan and Sepanski's approach to extremal $K$types.
Keywords:flag variety, symmetric subgroup Categories:14M15, 20G05 

62. CJM 2000 (vol 52 pp. 438)
 Wallach, N. R.; Willenbring, J.

On Some $q$Analogs of a Theorem of KostantRallis
In the first part of this paper generalizations of Hesselink's
$q$analog of Kostant's multiplicity formula for the action of a
semisimple Lie group on the polynomials on its Lie algebra are given
in the context of the KostantRallis theorem. They correspond to the
cases of real semisimple Lie groups with one conjugacy class of Cartan
subgroup. In the second part of the paper a $q$analog of the
KostantRallis theorem is given for the real group $\SL(4,\mathbb{R})$
(that is $\SO(4)$ acting on symmetric $4 \times 4$ matrices). This
example plays two roles. First it contrasts with the examples of the
first part. Second it has implications to the study of entanglement
of mixed 2 qubit states in quantum computation.
Categories:22E47, 20G05 

63. CJM 2000 (vol 52 pp. 197)
 Radjavi, Heydar

Sublinearity and Other Spectral Conditions on a Semigroup
Subadditivity, sublinearity, submultiplicativity, and other
conditions are considered for spectra of pairs of operators on a
Hilbert space. Sublinearity, for example, is a weakening of the
wellknown property~$L$ and means $\sigma(A+\lambda B) \subseteq
\sigma(A) + \lambda \sigma(B)$ for all scalars $\lambda$. The
effect of these conditions is examined on commutativity,
reducibility, and triangularizability of multiplicative semigroups
of operators. A sample result is that sublinearity of spectra
implies simultaneous triangularizability for a semigroup of compact
operators.
Categories:47A15, 47D03, 15A30, 20A20, 47A10, 47B10 

64. CJM 1999 (vol 51 pp. 1149)
 Cohen, A. M.; Cuypers, H.; Sterk, H.

Linear Groups Generated by Reflection Tori
A reflection is an invertible linear transformation of a vector
space fixing a given hyperplane, its axis, vectorwise and a given
complement to this hyperplane, its center, setwise. A reflection
torus is a onedimensional group generated by all reflections with
fixed axis and center.
In this paper we classify subgroups of general linear groups (in
arbitrary dimension and defined over arbitrary fields) generated by
reflection tori.
Categories:20Hxx, 20Gxx, 51A50 

65. CJM 1999 (vol 51 pp. 1307)
 Johnson, Norman W.; Weiss, Asia Ivić

Quadratic Integers and Coxeter Groups
Matrices whose entries belong to certain rings of algebraic
integers can be associated with discrete groups of transformations
of inversive $n$space or hyperbolic $(n+1)$space
$\mbox{H}^{n+1}$. For small $n$, these may be Coxeter groups,
generated by reflections, or certain subgroups whose generators
include direct isometries of $\mbox{H}^{n+1}$. We show how linear
fractional transformations over rings of rational and (real or
imaginary) quadratic integers are related to the symmetry groups of
regular tilings of the hyperbolic plane or 3space. New light is
shed on the properties of the rational modular group $\PSL_2
(\bbZ)$, the Gaussian modular (Picard) group $\PSL_2 (\bbZ[{\it
i}])$, and the Eisenstein modular group $\PSL_2 (\bbZ[\omega ])$.
Categories:11F06, 20F55, 20G20, 20H10, 22E40 

66. CJM 1999 (vol 51 pp. 1240)
67. CJM 1999 (vol 51 pp. 1226)
68. CJM 1999 (vol 51 pp. 1194)
 Lusztig, G.

Subregular Nilpotent Elements and Bases in $K$Theory
In this paper we describe a canonical basis for the equivariant
$K$theory (with respect to a $\bc^*$action) of the variety of
Borel subalgebras containing a subregular nilpotent element of a
simple complex Lie algebra of type $D$ or~$E$.
Category:20G99 

69. CJM 1999 (vol 51 pp. 1175)
 Lehrer, G. I.; Springer, T. A.

Reflection Subquotients of Unitary Reflection Groups
Let $G$ be a finite group generated by (pseudo) reflections in a
complex vector space and let $g$ be any linear transformation which
normalises $G$. In an earlier paper, the authors showed how to
associate with any maximal eigenspace of an element of the coset
$gG$, a subquotient of $G$ which acts as a reflection group on the
eigenspace. In this work, we address the questions of
irreducibility and the coexponents of this subquotient, as well as
centralisers in $G$ of certain elements of the coset. A criterion
is also given in terms of the invariant degrees of $G$ for an
integer to be regular for $G$. A key tool is the investigation of
extensions of invariant vector fields on the eigenspace, which
leads to some results and questions concerning the geometry of
intersections of invariant hypersurfaces.
Categories:51F15, 20H15, 20G40, 20F55, 14C17 

70. CJM 1999 (vol 51 pp. 881)
 Witherspoon, Sarah J.

The Representation Ring and the Centre of a Hopf Algebra
When $H$ is a finite dimensional, semisimple, almost cocommutative
Hopf algebra, we examine a table of characters which extends the
notion of the character table for a finite group. We obtain a
formula for the structure constants of the representation ring in
terms of values in the character table, and give the example of the
quantum double of a finite group. We give a basis of the centre of
$H$ which generalizes the conjugacy class sums of a finite group,
and express the class equation of $H$ in terms of this basis. We
show that the representation ring and the centre of $H$ are dual
character algebras (or signed hypergroups).
Categories:16W30, 20N20 

71. CJM 1999 (vol 51 pp. 658)
 Shumyatsky, Pavel

Nilpotency of Some Lie Algebras Associated with $p$Groups
Let $ L=L_0+L_1$ be a $\mathbb{Z}_2$graded Lie algebra over a
commutative ring with unity in which $2$ is invertible. Suppose
that $L_0$ is abelian and $L$ is generated by finitely many
homogeneous elements $a_1,\dots,a_k$ such that every commutator in
$a_1,\dots,a_k$ is adnilpotent. We prove that $L$ is nilpotent.
This implies that any periodic residually finite $2'$group $G$
admitting an involutory automorphism $\phi$ with $C_G(\phi)$
abelian is locally finite.
Categories:17B70, 20F50 

72. CJM 1998 (vol 50 pp. 1176)
 Dobson, Edward

Isomorphism problem for metacirculant graphs of order a product of distinct primes
In this paper, we solve the isomorphism problem for metacirculant
graphs of order $pq$ that are not circulant. To solve this problem,
we first extend Babai's characterization of the CIproperty to
nonCayley vertextransitive hypergraphs. Additionally, we find a
simple characterization of metacirculant Cayley graphs of order $pq$,
and exactly determine the full isomorphism classes of circulant graphs
of order $pq$.
Categories:05, 20 

73. CJM 1998 (vol 50 pp. 1007)
 Elder, G. Griffith

Galois module structure of ambiguous ideals in biquadratic extensions
Let $N/K$ be a biquadratic extension of algebraic number fields, and
$G=\Gal (N/K)$. Under a weak restriction on the ramification filtration
associated with each prime of $K$ above $2$, we explicitly describe the
$\bZ[G]$module structure of each ambiguous ideal of $N$. We find under
this restriction that in the representation of each ambiguous ideal as a
$\bZ[G]$module, the exponent (or multiplicity) of each indecomposable
module is determined by the invariants of ramification, alone.
For a given group, $G$, define ${\cal S}_G$ to be the set of
indecomposable $\bZ[G]$modules, ${\cal M}$, such that there
is an extension, $N/K$, for which $G\cong\Gal (N/K)$, and ${\cal M}$
is a $\bZ[G]$module summand of an ambiguous ideal of $N$. Can
${\cal S}_G$ ever be infinite? In this paper we answer this
question of Chinburg in the affirmative.
Keywords:Galois module structure, wild ramification Categories:11R33, 11S15, 20C32 

74. CJM 1998 (vol 50 pp. 829)
 Putcha, Mohan S.

Conjugacy classes and nilpotent variety of a reductive monoid
We continue in this paper our study of conjugacy classes
of a reductive monoid $M$. The main theorems establish a strong connection
with the BruhatRenner decomposition of $M$. We use our results to decompose
the variety $M_{\nil}$ of nilpotent elements of $M$ into irreducible components.
We also identify a class of nilpotent elements that we call standard and prove
that the number of conjugacy classes of standard nilpotent elements is always
finite.
Categories:20G99, 20M10, 14M99, 20F55 

75. CJM 1998 (vol 50 pp. 719)
 Göbel, Rüdiger; Shelah, Saharon

Indecomposable almost free modulesthe local case
Let $R$ be a countable, principal ideal domain which is not a field and
$A$ be a countable $R$algebra which is free as an $R$module. Then we
will construct an $\aleph_1$free $R$module $G$ of rank $\aleph_1$
with endomorphism algebra End$_RG = A$. Clearly the result does not
hold for fields. Recall that an $R$module is $\aleph_1$free if all
its countable submodules are free, a condition closely related to
Pontryagin's theorem. This result has many consequences, depending on
the algebra $A$ in use. For instance, if we choose $A = R$, then
clearly $G$ is an indecomposable `almost free' module. The existence of
such modules was unknown for rings with only finitely many primes like
$R = \hbox{\Bbbvii Z}_{(p)}$, the integers localized at some prime $p$. The result
complements a classical realization theorem of Corner's showing that
any such algebra is an endomorphism algebra of some torsionfree,
reduced $R$module $G$ of countable rank. Its proof is based on new
combinatorialalgebraic techniques related with what we call {\it rigid
treeelements\/} coming from a module generated over a forest of trees.
Keywords:indecomposable modules of local rings, $\aleph_1$free modules of rank $\aleph_1$, realizing rings as endomorphism rings Categories:20K20, 20K26, 20K30, 13C10 
