Expand all Collapse all | Results 76 - 100 of 107 |
76. CMB 2003 (vol 46 pp. 509)
Symmetries of Kirchberg Algebras Let $G_0$ and $G_1$ be countable abelian groups. Let $\gamma_i$ be an
automorphism of $G_i$ of order two. Then there exists a unital
Kirchberg algebra $A$ satisfying the Universal Coefficient Theorem and
with $[1_A] = 0$ in $K_0 (A)$, and an automorphism $\alpha \in
\Aut(A)$ of order two, such that $K_0 (A) \cong G_0$, such that $K_1
(A) \cong G_1$, and such that $\alpha_* \colon K_i (A) \to K_i (A)$ is
$\gamma_i$. As a consequence, we prove that every
$\mathbb{Z}_2$-graded countable module over the representation ring $R
(\mathbb{Z}_2)$ of $\mathbb{Z}_2$ is isomorphic to the equivariant
$K$-theory $K^{\mathbb{Z}_2} (A)$ for some action of $\mathbb{Z}_2$ on
a unital Kirchberg algebra~$A$.
Along the way, we prove that every not necessarily finitely generated
$\mathbb{Z} [\mathbb{Z}_2]$-module which is free as a
$\mathbb{Z}$-module has a direct sum decomposition with only three
kinds of summands, namely $\mathbb{Z} [\mathbb{Z}_2]$ itself and
$\mathbb{Z}$ on which the nontrivial element of $\mathbb{Z}_2$ acts
either trivially or by multiplication by $-1$.
Categories:20C10, 46L55, 19K99, 19L47, 46L40, 46L80 |
77. CMB 2003 (vol 46 pp. 332)
Some Questions about Semisimple Lie Groups Originating in Matrix Theory We generalize the well-known result that a square traceless complex
matrix is unitarily similar to a matrix with zero diagonal to
arbitrary connected semisimple complex Lie groups $G$ and their Lie
algebras $\mathfrak{g}$ under the action of a maximal compact subgroup
$K$ of $G$. We also introduce a natural partial order on
$\mathfrak{g}$: $x\le y$ if $f(K\cdot x) \subseteq f(K\cdot y)$ for
all $f\in \mathfrak{g}^*$, the complex dual of $\mathfrak{g}$. This
partial order is $K$-invariant and induces a partial order on the
orbit space $\mathfrak{g}/K$. We prove that, under some restrictions
on $\mathfrak{g}$, the set $f(K\cdot x)$ is star-shaped with respect
to the origin.
Categories:15A45, 20G20, 22E60 |
78. CMB 2003 (vol 46 pp. 268)
Group Cohomology and $L^p$-Cohomology of Finitely Generated Groups Let $G$ be a finitely generated, infinite group, let $p>1$, and let
$L^p(G)$ denote the Banach space $\{ \sum_{x\in G} a_xx \mid \sum_{x\in
G} |a_x |^p < \infty \}$. In this paper we will study the first
cohomology group of $G$ with coefficients in $L^p(G)$, and the first
reduced $L^p$-cohomology space of $G$. Most of our results will be for a
class of groups that contains all finitely generated, infinite nilpotent
groups.
Keywords:group cohomology, $L^p$-cohomology, central element of infinite order, harmonic function, continuous linear functional Categories:43A15, 20F65, 20F18 |
79. CMB 2003 (vol 46 pp. 310)
Second Order Dehn Functions of Asynchronously Automatic Groups Upper bounds of second order Dehn functions of asynchronously
automatic groups are obtained.
Keywords:second order Dehn function, combing, asynchronously automatic group Categories:20E06, 20F05, 57M05 |
80. CMB 2003 (vol 46 pp. 299)
A Basis of Bachmuth Type in the Commutator Subgroup of a Free Group We show here that the commutator subgroup of a free group of finite
rank poses a basis of Bachmuth's type.
Categories:20E05, 20F12, 20F05 |
81. CMB 2003 (vol 46 pp. 204)
Rationality and Orbit Closures Suppose we are given a finite-dimensional vector space $V$ equipped
with an $F$-rational action of a linearly algebraic group $G$, with
$F$ a characteristic zero field. We conjecture the following: to each
vector $v\in V(F)$ there corresponds a canonical $G(F)$-orbit of
semisimple vectors of $V$. In the case of the adjoint action, this
orbit is the $G(F)$-orbit of the semisimple part of $v$, so this
conjecture can be considered a generalization of the Jordan
decomposition. We prove some cases of the conjecture.
Categories:14L24, 20G15 |
82. CMB 2003 (vol 46 pp. 140)
An Explicit Cell Decomposition of the Wonderful Compactification of a Semisimple Algebraic Group We determine an explicit cell decomposition of the wonderful
compactification of a semi\-simple algebraic group. To do this we first
identify the $B\times B$-orbits using the generalized Bruhat
decomposition of a reductive monoid. From there we show how each cell
is made up from $B\times B$-orbits.
Categories:14L30, 14M17, 20M17 |
83. CMB 2003 (vol 46 pp. 122)
On Certain Finitely Generated Subgroups of Groups Which Split Define a group $G$ to be in the class $\mathcal{S}$ if for any
finitely generated subgroup $K$ of $G$ having the property that
there is a positive integer $n$ such that $g^n \in K$ for all
$g\in G$, $K$ has finite index in $G$. We show that a free
product with amalgamation $A*_C B$ and an $\HNN$ group $A *_C$ belong
to $\mathcal{S}$, if $C$ is in $\mathcal{S}$ and every subgroup of
$C$ is finitely generated.
Keywords:free product with amalgamation, $\HNN$ group, graph of groups, fundamental group Categories:20E06, 20E08, 57M07 |
84. CMB 2002 (vol 45 pp. 686)
An Aspect of Icosahedral Symmetry We embed the moduli space $Q$ of 5 points on the projective line
$S_5$-equivariantly into $\mathbb{P} (V)$, where $V$ is the
6-dimensional irreducible module of the symmetric group $S_5$. This
module splits with respect to the icosahedral group $A_5$ into the two
standard 3-dimensional representations. The resulting linear
projections of $Q$ relate the action of $A_5$ on $Q$ to those on the
regular icosahedron.
Categories:14L24, 20B25 |
85. CMB 2002 (vol 45 pp. 537)
86. CMB 2002 (vol 45 pp. 388)
AlgÃ¨bres simples centrales de degrÃ© 5 et $E_8$ As a consequence of a theorem of Rost-Springer, we establish that the
cyclicity problem for central simple algebra of degree~5 on fields
containg a fifth root of unity is equivalent to the study of
anisotropic elements of order 5 in the split group of type~$E_8$.
Keywords:algÃ¨bres simples centrales, cohomologie galoisienne Categories:16S35, 12G05, 20G15 |
87. CMB 2002 (vol 45 pp. 168)
Biquadratic Extensions with One Break We explicitly describe, in terms of indecomposable $\mathbb{Z}_2
[G]$-modules, the Galois module structure of ideals in totally
ramified biquadratic extensions of local number fields with only
one break in their ramification filtration. This paper completes
work begun in [Elder: Canad. J.~Math. (5) {\bf 50}(1998), 1007--1047].
Categories:11S15, 20C11 |
88. CMB 2002 (vol 45 pp. 294)
Modular Subgroups, Forms, Curves and Surfaces We study a class of subgroups of $\PSL_2 (\mathbb{Z})$ which can be
characterized in different ways, such as congruence groups, modular
forms, modular curves, elliptic surfaces, lattices and graphs.
Category:20H05 |
89. CMB 2001 (vol 44 pp. 385)
A Hypergraph with Commuting Partial Laplacians Let $F$ be a totally real number field and let $\GL_{n}$ be the
general linear group of rank $n$ over $F$. Let $\mathfrak{p}$
be a prime ideal of $F$ and $F_{\mathfrak{p}}$ the completion of $F$
with respect to the valuation induced by $\mathfrak{p}$. We will
consider a finite quotient of the affine building of the group
$\GL_{n}$ over the field $F_{\mathfrak{p}}$. We will view this object
as a hypergraph and find a set of commuting operators whose sum will
be the usual adjacency operator of the graph underlying the hypergraph.
Keywords:Hecke operators, buildings Categories:11F25, 20F32 |
90. CMB 2001 (vol 44 pp. 93)
Some Semigroup Laws in Groups A challenge by R.~Padmanabhan to prove by group theory the
commutativity of cancellative semigroups satisfying a particular
law has led to the proof of more general semigroup laws being
equivalent to quite simple ones.
Categories:20E10, 20M07 |
91. CMB 2001 (vol 44 pp. 27)
Normal Subloops in the Integral Loop Ring of an $\RA$ Loop We show that an $\RA$ loop has a torsion-free normal complement in the
loop of normalized units of its integral loop ring. We also
investigate whether an $\RA$ loop can be normal in its unit loop.
Over fields, this can never happen.
Categories:20N05, 17D05, 16S34, 16U60 |
92. CMB 2000 (vol 43 pp. 268)
Cockcroft Properties of Thompson's Group In a study of the word problem for groups, R.~J.~Thompson
considered a certain group $F$ of self-homeomorphisms of the Cantor
set and showed, among other things, that $F$ is finitely presented.
Using results of K.~S.~Brown and R.~Geoghegan, M.~N.~Dyer showed
that $F$ is the fundamental group of a finite two-complex $Z^2$
having Euler characteristic one and which is {\em Cockcroft}, in
the sense that each map of the two-sphere into $Z^2$ is
homologically trivial. We show that no proper covering complex of
$Z^2$ is Cockcroft. A general result on Cockcroft properties
implies that no proper regular covering complex of any finite
two-complex with fundamental group $F$ is Cockcroft.
Keywords:two-complex, covering space, Cockcroft two-complex, Thompson's group Categories:57M20, 20F38, 57M10, 20F34 |
93. CMB 2000 (vol 43 pp. 79)
Cyclotomic Schur Algebras and Blocks of Cyclic Defect An explicit classification is given of blocks of cyclic defect of
cyclotomic Schur algebras and of cyclotomic Hecke algebras, over
discrete valuation rings.
Categories:20G05, 20C20, 16G30, 17B37, 57M25 |
94. CMB 1999 (vol 42 pp. 335)
Cyclic Subgroup Separability of HNN-Extensions with Cyclic Associated Subgroups We derive a necessary and sufficient condition for HNN-extensions
of cyclic subgroup separable groups with cyclic associated
subgroups to be cyclic subgroup separable. Applying this, we
explicitly characterize the residual finiteness and the cyclic
subgroup separability of HNN-extensions of abelian groups with
cyclic associated subgroups. We also consider these residual
properties of HNN-extensions of nilpotent groups with cyclic
associated subgroups.
Keywords:HNN-extension, nilpotent groups, cyclic subgroup separable $(\pi_c)$, residually finite Categories:20E26, 20E06, 20F10 |
95. CMB 1999 (vol 42 pp. 298)
Semigroup Algebras and Maximal Orders We describe contracted semigroup algebras of Malcev nilpotent
semigroups that are prime Noetherian maximal orders.
Categories:16S36, 16H05, 20M25 |
96. CMB 1998 (vol 41 pp. 385)
Inequalities for Baer invariants of finite groups In this note we further our investigation of Baer invariants of
groups by obtaining, as consequences of an exact sequence of
A.~S.-T.~Lue, some numerical inequalities for their orders,
exponents, and generating sets. An interesting group theoretic
corollary is an explicit bound for $|\gamma_{c+1}(G)|$ given that
$G/Z_c(G)$ is a finite $p$-group with prescribed order and number
of generators.
Category:20C25 |
97. CMB 1998 (vol 41 pp. 488)
Remarks on certain metaplectic groups We study metaplectic coverings of the adelized group of a split
connected reductive group $G$ over a number field $F$. Assume its
derived group $G'$ is a simply connected simple Chevalley
group. The purpose is to provide some naturally defined sections
for the coverings with good properties which might be helpful when
we carry some explicit calculations in the theory of automorphic
forms on metaplectic groups. Specifically, we
\begin{enumerate}
\item construct metaplectic coverings of $G({\Bbb A})$ from those
of $G'({\Bbb A})$;
\item for any non-archimedean place $v$, show the section for a
covering of $G(F_{v})$ constructed from a Steinberg section is an
isomorphism, both algebraically and topologically in an open
subgroup of $G(F_{v})$;
\item define a global section which is a product of local sections
on a maximal torus, a unipotent subgroup and a set of
representatives for the Weyl group.
Categories:20G10, 11F75 |
98. CMB 1998 (vol 41 pp. 423)
Free products with amalgamation and $\lowercase{p}$-adic Lie groups Using the theory of $p$-adic Lie groups we give conditions for a
finitely generated group to admit a splitting as a non-trivial
free product with amalgamation. This can be viewed as an extension
of a theorem of Bass.
Category:20E06 |
99. CMB 1998 (vol 41 pp. 231)
The growth series of compact hyperbolic Coxeter groups with 4 and 5 generators The growth series of compact hyperbolic Coxeter groups with 4 and 5
generators are explicitly calculated. The assertions of J.~Cannon
and Ph.~Wagreich for the 4-generated groups, that the poles of the
growth series lie
on the unit circle, with the exception of a single real reciprocal pair of
poles, are verified. We also verify that for the 5-generated groups, this
phenomenon fails.
Categories:20F05, 20F55 |
100. CMB 1998 (vol 41 pp. 98)
Automorphisms of metabelian groups We investigate the problem of determining when $\IA (F_{n}({\bf A}_{m}{\bf A}))$
is finitely generated for all $n$ and $m$, with $n\geq 2$ and $m\neq 1$. If
$m$ is a nonsquare free integer then $\IA(F_{n}({\bf A}_{m}{\bf A}))$ is not
finitely generated for all $n$ and if $m$ is a square free integer then
$\IA(F_{n}({\bf A}_{m}{\bf A}))$ is finitely generated for all $n$, with
$n\neq 3$, and $\IA(F_{3}({\bf A}_{m}{\bf A}))$ is not finitely generated.
In case $m$ is square free, Bachmuth and Mochizuki claimed in ([7],
Problem 4) that $\TR({\bf A}_{m}{\bf A})$ is $1$ or $4$. We correct their
assertion by proving that $\TR({\bf A}_{m}{\bf A})=\infty $.
Category:20F28 |