26. CJM 2010 (vol 62 pp. 481)
 CasalsRuiz, Montserrat; Kazachkov, Ilya V.

Elements of Algebraic Geometry and the Positive Theory of Partially Commutative Groups
The first main result of the paper is a criterion for a partially commutative group $\mathbb G$ to be a domain. It allows us to reduce the study of algebraic sets over $\mathbb G$ to the study of irreducible algebraic sets, and reduce the elementary theory of $\mathbb G$ (of a coordinate group over $\mathbb G$) to the elementary theories of the direct factors of $\mathbb G$ (to the elementary theory of coordinate groups of irreducible algebraic sets). Then we establish normal forms for quantifierfree formulas over a nonabelian directly indecomposable partially commutative group $\mathbb H$. Analogously to the case of free groups, we introduce the notion of a generalised equation and prove that the positive theory of $\mathbb H$ has quantifier elimination and that arbitrary firstorder formulas lift from $\mathbb H$ to $\mathbb H\ast F$, where $F$ is a free group of finite rank. As a consequence, the positive theory of an arbitrary partially commutative group is decidable.
Categories:20F10, 03C10, 20F06 

27. CJM 2009 (vol 62 pp. 34)
 Campbell, Peter S.; Nevins, Monica

Branching Rules for Ramified Principal Series Representations of $\mathrm{GL}(3)$ over a $p$adic Field
We decompose the restriction of ramified principal series
representations of the $p$adic group $\mathrm{GL}(3,\mathrm{k})$ to its
maximal compact subgroup $K=\mathrm{GL}(3,R)$. Its decomposition is
dependent on the degree of ramification of the inducing characters and
can be characterized in terms of filtrations of the Iwahori subgroup
in $K$. We establish several irreducibility results and illustrate
the decomposition with some examples.
Keywords:principal series representations, branching rules, maximal compact subgroups, representations of $p$adic groups Categories:20G25, 20G05 

28. CJM 2009 (vol 61 pp. 950)
 Tange, Rudolf

Infinitesimal Invariants in a Function Algebra
Let $G$ be a reductive connected linear algebraic group
over an algebraically closed field of positive
characteristic and let $\g$ be its Lie algebra.
First we extend a wellknown result about the Picard group of a
semisimple group to reductive groups.
Then we prove that if the derived group is simply connected
and $\g$ satisfies a
mild condition, the algebra $K[G]^\g$ of regular functions
on $G$ that are invariant under the action of $\g$ derived
from the conjugation action is a unique factorisation domain.
Categories:20G15, 13F15 

29. CJM 2009 (vol 61 pp. 740)
 Caprace, PierreEmmanuel; Haglund, Frédéric

On Geometric Flats in the CAT(0) Realization of Coxeter Groups and Tits Buildings
Given a complete CAT(0) space $X$ endowed with a geometric action of a group $\Gamma$, it is known that if
$\Gamma$ contains a free abelian group of rank $n$, then $X$ contains a geometric flat of dimension $n$. We
prove the converse of this statement in the special case where $X$ is a convex subcomplex of the CAT(0)
realization of a Coxeter group $W$, and $\Gamma$ is a subgroup of $W$. In particular a convex cocompact subgroup
of a Coxeter group is Gromovhyperbolic if and only if it does not contain a free abelian group of rank 2. Our
result also provides an explicit control on geometric flats in the CAT(0) realization of arbitrary Tits
buildings.
Keywords:Coxeter group, flat rank, $\cat0$ space, building Categories:20F55, 51F15, 53C23, 20E42, 51E24 

30. CJM 2009 (vol 61 pp. 691)
 Yu, Xiaoxiang

Prehomogeneity on QuasiSplit Classical Groups and Poles of Intertwining Operators
Suppose that $P=MN$ is a maximal parabolic subgroup of a quasisplit,
connected, reductive classical group $G$ defined over a nonArchimedean
field and $A$ is the standard intertwining operator attached to a
tempered representation of $G$ induced from $M$. In this paper we
determine all the cases in which $\Lie(N)$ is
prehomogeneous under $\Ad(m)$ when $N$ is nonabelian, and give necessary
and sufficient conditions for $A$ to have a pole at $0$.
Categories:22E50, 20G05 

31. CJM 2009 (vol 61 pp. 708)
 Zelenyuk, Yevhen

Regular Homeomorphisms of Finite Order on Countable Spaces
We present a structure theorem for a broad class of homeomorphisms of
finite order on countable zero dimensional spaces. As applications we
show the following.
\begin{compactenum}[\rm(a)]
\item Every countable nondiscrete topological group not containing an
open Boolean subgroup can be partitioned into infinitely many dense
subsets.
\item If $G$ is a countably infinite Abelian group with finitely many
elements of order $2$ and $\beta G$ is the Stone\v Cech
compactification of $G$ as a discrete semigroup, then for every
idempotent $p\in\beta G\setminus\{0\}$, the subset
$\{p,p\}\subset\beta G$ generates algebraically the free product of
oneelement semigroups $\{p\}$ and~$\{p\}$.
\end{compactenum}
Keywords:Homeomorphism, homogeneous space, topological group, resolvability, Stone\v Cech compactification Categories:22A30, 54H11, 20M15, 54A05 

32. CJM 2008 (vol 60 pp. 1001)
33. CJM 2007 (vol 59 pp. 828)
 Ortner, Ronald; Woess, Wolfgang

NonBacktracking Random Walks and Cogrowth of Graphs
Let $X$ be a locally finite, connected graph without vertices of
degree $1$. Nonbacktracking random walk moves at each step with equal
probability to one of the ``forward'' neighbours of the actual state,
\emph{i.e.,} it does not go back along
the preceding edge to the preceding
state. This is not a Markov chain, but can be turned into a Markov
chain whose state space is the set of oriented edges of $X$. Thus we
obtain for infinite $X$ that the $n$step nonbacktracking transition
probabilities tend to zero, and we can also compute their limit when
$X$ is finite. This provides a short proof of old results concerning
cogrowth of groups, and makes the extension of that result to
arbitrary regular graphs rigorous. Even when $X$ is nonregular, but
\emph{small cycles are dense in} $X$, we show that the graph $X$ is
nonamenable if and only if the nonbacktracking $n$step transition
probabilities decay exponentially fast. This is a partial
generalization of the cogrowth criterion for regular graphs which
comprises the original cogrowth criterion for finitely generated
groups of Grigorchuk and Cohen.
Keywords:graph, oriented line grap, covering tree, random walk, cogrowth, amenability Categories:05C75, 60G50, 20F69 

34. CJM 2007 (vol 59 pp. 449)
 Badulescu, Alexandru Ioan

$\SL_n$, Orthogonality Relations and Transfer
Let $\pi$ be a square integrable representation of
$G'=\SL_n(D)$, with $D$ a central division algebra of finite dimension
over a local field $F$ \emph{of nonzero characteristic}. We prove
that, on the elliptic set, the character of $\pi$ equals the complex
conjugate of the orbital integral of one of the pseudocoefficients
of~$\pi$. We prove also the orthogonality relations for characters of
square integrable representations of $G'$. We prove the stable
transfer of orbital integrals between $\SL_n(F)$ and its inner forms.
Category:20G05 

35. CJM 2007 (vol 59 pp. 296)
 Chein, Orin; Goodaire, Edgar G.

Bol Loops of Nilpotence Class Two
Call a nonMoufang Bol loop \emph{minimally nonMoufang}
if every proper subloop is Moufang and
\emph{minimally nonassociative} if every proper subloop is
associative. We prove that these concepts are
the same for Bol loops which are nilpotent of
class two and in which certain associators square to $1$.
In the process, we derive many commutator and associator identities
which hold in such loops.
Keywords:Bol loop, Moufang loop, nilpotent, commutator, associator, minimally nonassociative Category:20N05 

36. CJM 2007 (vol 59 pp. 418)
 Stoimenow, A.

On Cabled Knots and Vassiliev Invariants (Not) Contained in Knot Polynomials
It is known that the BrandtLickorishMillettHo polynomial $Q$
contains Casson's knot invariant. Whether there are (essentially)
other Vassiliev knot invariants obtainable from $Q$ is an open
problem. We show that this is not so up to degree $9$. We also
give the (apparently) first examples of knots not distinguished
by 2cable HOMFLY polynomials which are not mutants. Our calculations
provide evidence of a negative answer to the question whether Vassiliev
knot invariants of degree $d \le 10$ are determined by the HOMFLY and
Kauffman polynomials and their 2cables, and for the existence of
algebras of such Vassiliev invariants not isomorphic to the algebras
of their weight systems.
Categories:57M25, 57M27, 20F36, 57M50 

37. CJM 2006 (vol 58 pp. 1144)
 Hamana, Masamichi

Partial $*$Automorphisms, Normalizers, and Submodules in Monotone Complete $C^*$Algebras
For monotone complete $C^*$algebras
$A\subset B$ with $A$ contained in $B$ as a monotone closed
$C^*$subalgebra, the relation $X = AsA$
gives a bijection between the set of all
monotone closed linear subspaces $X$ of $B$ such that
$AX + XA \subset X$
and
$XX^* + X^*X \subset A$
and a set of certain partial
isometries $s$ in the ``normalizer" of $A$ in $B$,
and similarly for the map $s \mapsto \Ad s$
between the latter set and a set of certain ``partial $*$automorphisms"
of $A$.
We introduce natural inverse semigroup
structures in the set of such $X$'s and the set of
partial $*$automorphisms of $A$, modulo a certain relation, so that
the composition of these maps induces an inverse semigroup
homomorphism between them.
For a large enough $B$ the homomorphism becomes surjective and
all the partial $*$automorphisms of
$A$ are realized via partial isometries in $B$.
In particular, the inverse semigroup associated with
a type ${\rm II}_1$ von Neumann factor,
modulo the outer automorphism group,
can be viewed as the fundamental group of the factor.
We also consider the $C^*$algebra version of these results.
Categories:46L05, 46L08, 46L40, 20M18 

38. CJM 2006 (vol 58 pp. 897)
 Courtès, François

Distributions invariantes sur les groupes rÃ©ductifs quasidÃ©ployÃ©s
Soit $F$ un corps local non archim\'edien, et $G$ le groupe des
$F$points d'un groupe r\'eductif connexe quasid\'eploy\'e d\'efini sur $F$.
Dans cet article, on s'int\'eresse aux distributions sur $G$ invariantes
par conjugaison, et \`a l'espace de leurs restrictions \`a l'alg\`ebre de
Hecke $\mathcal{H}$ des fonctions sur $G$ \`a support compact
biinvariantes par un sousgroupe d'Iwahori $I$ donn\'e. On montre tout
d'abord que les valeurs d'une telle distribution sur $\mathcal{H}$
sont enti\`erement d\'etermin\'ees par sa restriction au sousespace de
dimension finie des \'el\'ements de $\mathcal{H}$ \`a support dans la
r\'eunion des sousgroupes parahoriques de $G$ contenant $I$. On utilise
ensuite cette propri\'et\'e pour montrer, moyennant certaines conditions
sur $G$, que cet espace est engendr\'e d'une part par certaines
int\'egrales orbitales semisimples, d'autre part par les int\'egrales
orbitales unipotentes, en montrant tout d'abord des r\'esultats
analogues sur les groupes finis.
Keywords:reductive $p$adic groups, orbital integrals, invariant distributions Categories:22E35, 20G40 

39. CJM 2006 (vol 58 pp. 23)
 DabbaghianAbdoly, Vahid

Constructing Representations of Finite Simple Groups and Covers
Let $G$ be a finite group and $\chi$ be an irreducible character of $G$. An efficient
and simple method to construct representations of finite groups is applicable
whenever $G$ has a subgroup $H$ such that $\chi_H$
has a linear constituent with multiplicity $1$.
In this paper we show (with a few exceptions) that if $G$
is a simple group or a covering group of a simple group and
$\chi$ is an irreducible character of $G$ of degree less than 32,
then there exists a subgroup $H$ (often a Sylow subgroup) of $G$
such that $\chi_H$ has a linear constituent with multiplicity $1$.
Keywords:group representations, simple groups, central covers, irreducible representations Categories:20C40, 20C15 

40. CJM 2005 (vol 57 pp. 1056)
 Ozawa, Narutaka; Rieffel, Marc A.

Hyperbolic Group $C^*$Algebras and FreeProduct $C^*$Algebras as Compact Quantum Metric Spaces
Let $\ell$ be a length function on a group $G$, and let $M_{\ell}$
denote the
operator of pointwise multiplication by $\ell$ on $\bell^2(G)$.
Following Connes,
$M_{\ell}$ can be used as a ``Dirac'' operator for $C_r^*(G)$. It defines a
Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of
$C_r^*(G)$. We show that if $G$ is a hyperbolic group and if $\ell$ is
a wordlength function on $G$, then the topology from this metric
coincides with the
weak$*$ topology (our definition of a ``compact quantum metric
space''). We show that a convenient framework is that of filtered
$C^*$algebras which satisfy a suitable ``Haageruptype'' condition. We
also use this
framework to prove an analogous fact for certain reduced
free products of $C^*$algebras.
Categories:46L87, 20F67, 46L09 

41. CJM 2005 (vol 57 pp. 648)
 Nevins, Monica

Branching Rules for Principal Series Representations of $SL(2)$ over a $p$adic Field
We explicitly describe the decomposition into irreducibles of
the restriction of the principal
series representations of $SL(2,k)$, for $k$ a $p$adic field,
to each of its two maximal compact subgroups (up to conjugacy).
We identify these irreducible subrepresentations in the
Kirillovtype classification
of Shalika. We go on to explicitly describe the decomposition
of the reducible principal series of $SL(2,k)$ in terms of the
restrictions of its irreducible constituents to a maximal compact
subgroup.
Keywords:representations of $p$adic groups, $p$adic integers, orbit method, $K$types Categories:20G25, 22E35, 20H25 

42. CJM 2005 (vol 57 pp. 416)
 Wise, Daniel T.

Approximating Flats by Periodic Flats in \\CAT(0) Square Complexes
We investigate the problem of whether every immersed flat plane in a
nonpositively curved square complex is the limit of periodic flat
planes. Using a branched cover, we reduce the problem to the case of
$\V$complexes. We solve the problem for malnormal and cyclonormal
$\V$complexes. We also solve the problem for complete square
complexes using a different approach. We give an application towards
deciding whether the elements of fundamental groups of the spaces we
study have commuting powers. We note a connection between the flat
approximation problem and subgroup separability.
Keywords:CAT(0), periodic flat planes Categories:20F67, 20F06 

43. CJM 2004 (vol 56 pp. 945)
 Helminck, Aloysius G.; Schwarz, Gerald W.

Smoothness of Quotients Associated \\With a Pair of Commuting Involutions
Let $\sigma$, $\theta$ be commuting involutions of the connected semisimple
algebraic group $G$ where $\sigma$, $\theta$ and $G$ are defined over
an algebraically closed field $\k$, $\Char \k=0$. Let $H:=G^\sigma$
and $K:=G^\theta$ be the fixed point groups. We have an action
$(H\times K)\times G\to G$, where $((h,k),g)\mapsto hgk\inv$, $h\in
H$, $k\in K$, $g\in G$. Let $\quot G{(H\times K)}$ denote the
categorical quotient $\Spec \O(G)^{H\times K}$. We determine when this
quotient is smooth. Our results are a generalization of those of
Steinberg \cite{Steinberg75}, Pittie \cite{Pittie72} and Richardson
\cite{Rich82b} in the symmetric case where $\sigma=\theta$ and $H=K$.
Categories:20G15, 20G20, 22E15, 22E46 

44. CJM 2004 (vol 56 pp. 871)
 Schocker, Manfred

Lie Elements and Knuth Relations
A coplactic class in the symmetric group $\Sym_n$ consists of all
permutations in $\Sym_n$ with a given Schensted $Q$symbol, and may
be described in terms of local relations introduced by Knuth. Any
Lie element in the group algebra of $\Sym_n$ which is constant on
coplactic classes is already constant on descent classes. As a
consequence, the intersection of the Lie convolution algebra
introduced by Patras and Reutenauer and the coplactic algebra
introduced by Poirier and Reutenauer is the direct sum of all
Solomon descent algebras.
Keywords:symmetric group, descent set, coplactic relation, Hopf algebra,, convolution product Categories:17B01, 05E10, 20C30, 16W30 

45. CJM 2004 (vol 56 pp. 246)
 Bonnafé, Cédric

ÃlÃ©ments unipotents rÃ©guliers des sousgroupes de Levi
We investigate the structure of the centralizer of a regular unipotent element
of a Levi subgroup of a reductive group. We also investigate the structure of
the group of components of this centralizer in relation with the notion of
cuspidal local system defined by Lusztig. We determine its unipotent radical,
we prove that it admits a Levi complement, and we get some properties on its Weyl
group. As an application, we prove some results which were announced in previous
paper on regular unipotent elements.
Nous \'etudions la structure du centralisateur d'un \'el\'ement unipotent
r\'egulier d'un sousgroupe de Levi d'un groupe r\'eductif, ainsi que la structure
du groupe des composantes de ce centralisateur en relation avec la notion de
syst\`eme local cuspidal d\'efinie par Lusztig. Nous d\'eterminons son radical
unipotent, montrons l'existence d'un compl\'ement de Levi et \'etudions la
structure de son groupe de Weyl. Comme application, nous d\'emontrons des
r\'esultats qui \'etaient annonc\'es dans un pr\'ec\'edent article de l'auteur
sur les \'el\'ements unipotents r\'eguliers.
Category:20G 

46. 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 

47. 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 

48. 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 

49. CJM 2002 (vol 54 pp. 1229)
50. CJM 2002 (vol 54 pp. 970)