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1,300 |
Efficient representation of perm groups
|
math.GR
|
This note presents an elementary version of Sims's algorithm for computing
strong generators of a given perm group, together with a proof of correctness
and some notes about appropriate low-level data structures. Upper and lower
bounds on the running time are also obtained. (Following a suggestion of
Vaughan Pratt, we adopt the convention that perm $=$ permutation, perhaps
thereby saving millions of syllables in future research.)
|
math
|
1,301 |
The 1-, 2-, and 3-characters determine a group
|
math.GR
|
A set of invariants for a finite group is described. These arise naturally
from Frobenius' early work on the group determinant and provide an answer to a
question of Brauer. Whereas it is well known that the ordinary character table
of a group does not determine the group uniquely, it is a consequence of the
results presented here that a group is determined uniquely by its
``3-character'' table.
|
math
|
1,302 |
On the Burnside problem on periodic groups
|
math.GR
|
It is proved that the free $m$-generated Burnside groups $\Bbb{B}(m,n)$ of
exponent $n$ are infinite provided that $m>1$, $n\ge2^{48}$.
|
math
|
1,303 |
Musings on Magnus
|
math.GR
|
The object of this paper is to describe a simple method for proving that
certain groups are residually torsion-free nilpotent, to describe some new
parafree groups and to raise some new problems in honour of the memory of
Wilhelm Magnus.
|
math
|
1,304 |
(p,q,r)-Generations for Janko Groups $J_1$ and $J_2$
|
math.GR
|
No abstract is available
|
math
|
1,305 |
Almost Convex Groups and the Eight Geometries
|
math.GR
|
If $M$ is a closed Nil geometry 3-manifold then $\pi_1(M)$ is almost convex
with respect to a fairly simple ``geometric'' generating set. If $G$ is a
central extension or a ${\Bbb Z}$-extension of a word hyperbolic group, then
$G$ is also almost convex with respect to some generating set. Combining these
with previously known results shows that if $M$ is a closed 3-manifold with one
of Thurston's eight geometries, $\pi_1(M)$ is almost convex with respect to
some generating set if and only if the geometry in question is not Sol.
|
math
|
1,306 |
Sur un generalisation del notion de producto libere amalgamate de gruppos
|
math.GR
|
In ``A remark about the description of free products of groups'', Proc.
Cambgridge Philos. Soc 62(1966), io ha studite lo que occurre in le
circumstantia que un gruppo $G$ ha un subensemble $P$ tal que tote elemento de
$G$ es representabile unicamente per un verbo reducite in $P$. Il eveni que tal
$P$ es multo como un producto libere. Que occurre quando le representation per
verbo reducite es unic solmente modulo le sorta de equivalentia que interveni
in le theoria del productos libere amalgamate? In iste articulo, io determina
le structura internal del subensemble $P$ (io los appella ``pregruppos''), e
prova, sequente le methodo de van der Waerden, que su gruppo universal ha le
proprietate desiderate. Multe interessante exemplos pote esser trovate; tote
semble simile aliquanto al productos libere amalgamate; sed il es nulle simple
maniera de construer los omne ex ordinari tal productos.
|
math
|
1,307 |
A note on Context Sensitive languages and Word Problems
|
math.GR
|
Anisimov and Seifert show that a group has a regular word problem ifand only
if it is finite. Muller and Schupp (together with Dunwoody's accessibility
result) show that a group has context free word problem if and only if it is
virtually free. In this note, we exhibit a class of groups where the word
problem is as close as possible to being a context sensitive language. This
class includes the automatic groups and is closed under passing to finitely
generated subgroups. Consequently, it is quite large, including many groups
which are not finitely presented.
|
math
|
1,308 |
Automatic structures and boundaries for graphs of groups
|
math.GR
|
We study the synchronous and asynchronous automatic structures on the
fundamental group of a graph of groups in which each edge group is finite. Up
to a natural equivalence relation, the set of biautomatic structures on such a
graph product bijects to the product of the sets of biautomatic structures on
the vertex groups. The set of automatic structures is much richer. Indeed, it
is dense in the infinite product of the sets of automatic structures of all
conjugates of the vertex groups. We classify these structures by a class of
labelled graphs which ``mimic" the underlying graph of the graph of groups.
Analogous statements hold for asynchronous automatic structures. We also
discuss the boundaries of these structures.
|
math
|
1,309 |
The Bieri-Neumann-Strebel invariants for graph groups
|
math.GR
|
Given a finite simplicial graph ${\cal G}$, the graph group $G{\cal G}$" is
the group with generators in one-to-one correspondence with the vertices of
${\cal G}$ and with relations stating two generators commute if their
associated vertices are adjacent in ${\cal G}$. The Bieri-Neumann-Strebel
invariant can be explicitly described in terms of the original graph ${\cal G}$
and hence there is an explicit description of the distribution of finitely
generated normal subgroups of $G{\cal G}$ with abelian quotient. We construct
Eilenberg-MacLane spaces for graph groups and find partial extensions of this
work to the higher dimensional invariants.
|
math
|
1,310 |
Unions of Cockroft two-complexes
|
math.GR
|
A combinatorial group-theoretic hypothesis is presented that serves as a
necessary and sufficient condition for a union of connected Cockcroft
two-complexes to be Cockcroft. This hypothesis has a component that can be
expressed in terms of the second homology of groups. The hypothesis is applied
to the study of the third homology of groups given by generators and relators.
|
math
|
1,311 |
Cogrowth and essentiality in groups and algebras
|
math.GR
|
The cogrowth of a subgroup is defined as the growth of a set of coset
representatives which are of minimal length. A subgroup is essential if it
intersects non-trivially every non-trivial subgroup. The main result of this
paper is that every function $f:{\Bbb N}\cup \{0\}\rightarrow {\Bbb N}$ which
is strictly increasing, but at most exponential, is equivalent to a cogrowth
function of an essential subgroup of infinite index of the free group of rank
two. This class of functions properly contains the class of growth functions of
groups. The notions of growth and cogrowth of right ideals in algebras are
introduced. We show that when the algebra is without zero divisors then every
right ideal, whose cogrowth is less than that of the algebra, is essential.
|
math
|
1,312 |
Commutators as Powers in Free Products of Groups
|
math.GR
|
The ways in which a nontrivial commutator can be a proper power in a free
product of groups are identified.
|
math
|
1,313 |
Products of Commutators and Products of Squares in a Free Group
|
math.GR
|
A classification of the ways in which an element of a free group can be
expressed as a product of commutators or as a product of squares is given. This
is then applied to some particular classes of elements. Finally, a question
about expressing a commutator as a product of squares is addressed.
|
math
|
1,314 |
Projective resolutions for graph products
|
math.GR
|
Let $\Gamma$ be a finite graph together with a group $G_v$ at each vertex
$v$. The graph product $G(\Gamma)$ is obtained from the free product of all
$G_v$ by factoring out by the normal subgroup generated by $\{g^{-1}h^{-1}gh;
g\in G_v, h\in G_w\}$ for all adjacent $v,w$. In this note we construct a
projective resolution for $G(\Gamma)$ given projective resolutions for each
$G_v$, and obtain some applications.
|
math
|
1,315 |
Isoperimetric functions for graph products
|
math.GR
|
Let $\Gamma$ be a finite graph, and for each vertex $i$ let $G_i$ be a
finitely presented group. Let $G$ be the graph product of the $G_i$. That is,
$G$ is the group obtained from the free product of the $G_i$ by factoring out
by the smallest normal subgroup containing all $[g,h]$ where $g\in G_i$ and
$h\in G_j$ and there is an edge joining i and j . We show that $G$ has an
isoperimetric function of degree $k\ge 2$ (or an exponential isoperimetric
function) if each vertex group has such an isoperimetric function.
|
math
|
1,316 |
A bicombing that implies a sub-exponential isoperimetric inequality
|
math.GR
|
The idea of applying isoperimetric functions to group theory is due to
M.Gromov. We introduce the concept of a ``bicombing of narrow shape'' which
generalizes the usual notion of bicombing. Our bicombing is related to but
different from the combings defined by M. Bridson. If the Cayley graph of a
group with respect to a given set of generators admits a bicombing of narrow
shape then the group is finitely presented and satisfies a sub-exponential
isoperimetric inequality, as well as a polynomial isodiametric inequality. We
give an infinite class of examples which are not bicombable in the usual sense
but admit bicombings of narrow shape.
|
math
|
1,317 |
The Geometry of Cycles in the Cayley Diagram of a Group
|
math.GR
|
A study of triangulations of cycles in the Cayley diagrams of finitely
generated groups leads to a new geometric characterization of hyperbolic
groups.
|
math
|
1,318 |
Automatic structures, rational growth and geometrically finite hyperbolic groups
|
math.GR
|
We show that the set $SA(G)$ of equivalence classes of synchronously
automatic structures on a geometrically finite hyperbolic group $G$ is dense in
the product of the sets $SA(P)$ over all maximal parabolic subgroups $P$. The
set $BSA(G)$ of equivalence classes of biautomatic structures on $G$ is
isomorphic to the product of the sets $BSA(P)$ over the cusps (conjugacy
classes of maximal parabolic subgroups) of $G$. Each maximal parabolic $P$ is a
virtually abelian group, so $SA(P)$ and $BSA(P)$ were computed in ``Equivalent
automatic structures and their boundaries'' by M.Shapiro and W.Neumann, Intern.
J. of Alg. Comp. 2 (1992) We show that any geometrically finite hyperbolic
group has a generating set for which the full language of geodesics for $G$ is
regular. Moreover, the growth function of $G$ with respect to this generating
set is rational. We also determine which automatic structures on such a group
are equivalent to geodesic ones. Not all are, though all biautomatic structures
are.
|
math
|
1,319 |
Hyperbolic buildings, affine buildings and automatic groups
|
math.GR
|
We see that a building whose Coxeter group is hyperbolic is itself
hyperbolic. Thus any finitely generated group acting co-compactly on such a
building is hyperbolic, hence automatic. We turn our attention to affine
buildings and consider a group $\Gamma$ which acts simply transitively and in a
``type-rotating'' way on the vertices of a locally finite thick building of
type $\tilde A_n$. We show that $\Gamma$ is biautomatic, using a presentation
of $\Gamma$ and unique normal form for each element of $\Gamma$, as described
in ``Groups acting simply transitively on the vertices of a building of type
$\tilde A_n$'' by D.I. Cartwright, to appear, Proceedings of the 1993 Como
conference ``Groups of Lie type and their geometries''.
|
math
|
1,320 |
Central quotients of biautomatic groups
|
math.GR
|
The quotient of a biautomatic group by a subgroup of the center is shown to
be biautomatic. The main tool used is the Neumann-Shapiro triangulation of
$S^{n-1}$, associated to a biautomatic structure on ${\Bbb Z}^n$. As an
application, direct factors of biautomatic groups are shown to be biautomatic.
|
math
|
1,321 |
Coset enumeration strategies
|
math.GR
|
A primary reference on computer implementation of coset enumeration
procedures is a 1973 paper of Cannon, Dimino, Havas and Watson. Programs and
techniques described there are updated in this paper. Improved coset definition
strategies, space saving techniques and advice for obtaining improved
performance are included. New coset definition strategies for Felsch-type
methods give substantial reductions in total cosets defined for some
pathological enumerations. Significant time savings are achieved for coset
enumeration procedures in general. Statistics on performance are presented,
both in terms of time and in terms of maximum and total cosets defined for
selected enumerations.
|
math
|
1,322 |
Algorithms for groups
|
math.GR
|
Group theory is a particularly fertile field for the design of practical
algorithms. Algorithms have been developed across the various branches of the
subject and they find wide application. Because of its relative maturity,
computational group theory may be used to gain insight into the general
structure of algebraic algorithms. This paper examines the basic ideas behind
some of the more important algorithms for finitely presented groups and
permutation groups, and surveys recent developments in these fields.
|
math
|
1,323 |
Applications of substring searching to group presentations
|
math.GR
|
An important way for describing groups is by finite presentations. Large
presentations arise in practice which are poorly suited for either human or
computer use. Presentation simplification processes which take bad
presentations and produce good presentations have been developed. Substantial
use is made of substring searching and appropriate techniques for this context
are described. Effective use is made of signatures and change flags. Change
flags are shown to be the most beneficial of the methods tested here, with very
significant performance improvement. Experimental performance figures are
given.
|
math
|
1,324 |
Recognizing badly presented Z-modules
|
math.GR
|
Finitely generated Z-modules have canonical decompositions. When such modules
are given in a finitely presented form there is a classical algorithm for
computing a canonical decomposition. This is the algorithm for computing the
Smith normal form of an integer matrix. We discuss algorithms for Smith normal
form computation, and present practical algorithms which give excellent
performance for modules arising from badly presented abelian groups. We
investigate such issues as congruential techniques, sparsity considerations,
pivoting strategies for Gauss-Jordan elimination, lattice basis reduction and
computational complexity. Our results, which are primarily empirical, show
dramatically improved performance on previous methods.
|
math
|
1,325 |
A new problem in string searching
|
math.GR
|
We describe a substring search problem that arises in group presentation
simplification processes. We suggest a two-level searching model: skip and
match levels. We give two timestamp algorithms which skip searching parts of
the text where there are no matches at all and prove their correctness. At the
match level, we consider Harrison signature, Karp-Rabin fingerprint, Bloom
filter and automata based matching algorithms and present experimental
performance figures.
|
math
|
1,326 |
Applications of computational tools for finitely presented groups
|
math.GR
|
Computer based techniques for recognizing finitely presented groups are quite
powerful. Tools available for this purpose are outlined. They are available
both in stand-alone programs and in more comprehensive systems. A general
computational approach for investigating finitely presented groups by way of
quotients and subgroups is described and examples are presented. The techniques
can provide detailed information about group structure. Under suitable
circumstances a finitely presented group can be shown to be soluble and its
complete derived series can be determined, using what is in effect a soluble
quotient algorithm.
|
math
|
1,327 |
The flag-transitive tilde and Petersen-type geometries are all known
|
math.GR
|
We announce the classification of two related classes of flag-transitive
geometries. There is an infinite family of such geometries, related to the
nonsplit extensions $3^{[{n\atop 2}]_{_2}}\cdot \SP_{2n}(2)$, and twelve
sporadic examples coming from the simple groups $M_{22}$, $M_{23}$, $M_{24}$,
$He$, $Co_1$, $Co_2$, $J_4$, $BM$, $M$ and the nonsplit extensions $3\cdot
M_{22}$, $3^{23}\cdot Co_2$, and $3^{4371}\cdot BM$.
|
math
|
1,328 |
Regular Cocycles and Biautomatic Structures
|
math.GR
|
Let $E$ be a virtually central extension of the group $G$ by a finitely
generated abelian group $A$. We show that $E$ carries a biautomatic structure
if and only if $G$ has a biautomatic structure $L$ for which the cohomology
class of the extension is represented by an $L$-regular cocycle. Moreover, a
cohomology class is $L$-regular if some multiple of it is or if its restriction
to some finite index subgroup is.
We also show that the entire second cohomology of a Fuchsian group is
regular, so any virtually central extension is biautomatic. In particular, if
the fundamental group of a Seifert fibered 3-manifold is not virtually
nilpotent then it is biautomatic. ECHLPT had shown automaticity in this case
and in an unpublished 1992 preprint Gersten constructed a biautomatic structure
for circle bundles over hyperbolic surfaces and asked if the same could be done
for these Seifert fibered 3-manifolds.
|
math
|
1,329 |
When Schrier transversals grow wild
|
math.GR
|
Schreier formula for the rank of a subgroup of finite index of a finitely
generated free group $F$ is generalized to an arbitrary (even infinitely
generated) subgroup $H$ through the Schreier transversals of $H$ in $F$. The
rank formula may also be expressed in terms of the cogrowth of $H$. We
introduce the rank-growth function $rk_H(i)$ of a subgroup $H$ of a finitely
generated free group $F$. $rk_H(i)$ is defined to be the rank of the subgroup
of $H$ generated by elements of length less than or equal to $i$ (with respect
to the generators of $F$), and it equals the rank of the fundamental group of
the subgraph of the cosets graph of $H$, which consists of the paths starting
at $1$ that are of length $\leq i$. When $H$ is supnormal, i.e. contains a
non-trivial normal subgroup of $F$, we show that its rank-growth is equivalent
to the cogrowth of $H$. A special case of this is the known result that a
supnormal subgroup of $F$ is of finite index if and only if it is finitely
generated. In particular, when $H$ is normal then the growth of the group
$G=F/H$ is equivalent to the rank-growth of $H$. A Schreier transversal forms a
spanning tree of the cosets graph of $H$, and thus its topological structure is
of a contractible spanning subcomplex of a simplicial complex. The
$d$-dimensional simplicial complexes that contain contractible spanning
subcomplexes have the homotopy type of a bouquet of $r$ $d$-spheres. When these
complexes are also $n$-regular then $r$ can be computed by generalizing the
rank formula (which applies to Schreier transversals) to higher dimensions.
|
math
|
1,330 |
The normalized cyclomatic quotient associated with presentations of finitely generated groups
|
math.GR
|
Given the Cayley graph of a finitely generated group $G$, with respect to a
presentation $G^{\alpha}$ with $n$ generators, the quotient of the rank of the
fundamental group of subgraphs of the Cayley graph by the cardinality of the
set of vertices of the subgraphs gives rise to the definition of the normalized
cyclomatic quotient $\Xi (G^{\alpha})$. The asymptotic behavior of this
quotient is similar to the asymptotic behavior of the quotient of the
cardinality of the boundary of the subgraph by the cardinality of the subgraph.
Using Følner's criterion for amenability one gets that $\Xi (G^{\alpha})$
vanishes for infinite groups if and only if they are amenable. When $G$ is
finite then $\Xi (G^{\alpha})=1/|G|$, where $|G|$'> is the cardinality of $G$,
and when $G$ is non-amenable then $1-n\leq\Xi (G^{\alpha})\le 0$, with $\Xi
(G^{\alpha})=1-n$ if and only if $G$ is free of rank $n$. Thus we see that on
special cases $\Xi (G^{\alpha})$ takes the values of the Euler characteristic
of $G$. Most of the paper is concerned with formulae for the value of $\Xi
(G^{\alpha})$ with respect to that of subgroups and factor groups, and with
respect to the decomposition of the group into direct product and free product.
Some of the formulae and bounds we get for $\Xi (G^{\alpha})$ are similar to
those given for the spectral radius of symmetric random walks on the graph of
$G^{\alpha}$, but this is not always the case. In the last section of the paper
we define and touch very briefly the balanced cyclomatic quotient, which is
defined on concentric balls in the graph and is related to the growth of $G$.
|
math
|
1,331 |
On finite induced crossed modules and the homotopy 2-type of mapping cones
|
math.GR
|
Results on the finiteness of induced crossed modules are proved both
algebraically and topologically. Using the Van Kampen type theorem for the
fundamental crossed module, applications are given to the 2-types of mapping
cones of classifying spaces of groups. Calculations of the cohomology classes
of some finite crossed modules are given, using crossed complex methods.
|
math
|
1,332 |
The second bounded cohomology of a group with infinitely many ends
|
math.GR
|
We study the second bounded cohomology of an amalgamated free product of
groups, and an HNN extension of a group. As an application, we have a group
with infinitely many ends has infinite dimensional second bounded cohomology.
|
math
|
1,333 |
Detecting quasiconvexity: algorithmic aspects
|
math.GR
|
The main result of this paper states that for any group $G$ with an automatic
structure $L$ with unique representatives one can construct a uniform partial
algorithm which detects $L$-rational subgroups and gives their preimages in
$L$. This provides a practical, not just theoretical, procedure for solving the
occurrence problem for such subgroups.
|
math
|
1,334 |
Quasiconvexity and Amalgams
|
math.GR
|
We obtain a criterion for quasiconvexity of a subgroup of an amalgamated free
product of two word hyperbolic groups along a virtually cyclic subgroup. The
result provides a method of constructing new word hyperbolic group in class
(Q), that is such that all their finitely generated subgroups are quasiconvex.
It is known that free groups, hyperbolic surface groups and most 3-dimensional
Kleinian groups have property (Q). We also give some applications of our
results to one-relator groups and exponential groups.
|
math
|
1,335 |
An example of a non-quasiconvex subgroup of a word hyperbolic group with exotic limit set
|
math.GR
|
We construct an example of a torsion free freely indecomposable finitely
presented non-quasiconvex subgroup $H$ of a word hyperbolic group $G$ such that
the limit set of $H$ is not the limit set of a quasiconvex subgroup of $G$. In
particular, this gives a counterexample to the conjecture of G.Swarup that a
finitely presented one-ended subgroup of a word hyperbolic group is quasiconvex
if and only if it has finite index in its virtual normalizer.
|
math
|
1,336 |
Central Extensions of Word Hyperbolic Groups
|
math.GR
|
Thurston has claimed (unpublished) that central extensions of word hyperbolic
groups by finitely generated abelian groups are automatic. We show that they
are in fact biautomatic. Further, we show that every 2-dimensional cohomology
class on a word hyperbolic group can be represented by a bounded 2-cocycle.
This lends weight to the claim of Gromov that for a word hyperbolic group, the
cohomology in every dimension is bounded.
|
math
|
1,337 |
The Warwick Automatic Groups Software
|
math.GR
|
This paper provides a description of the algorithms employed by the Warwick
AUTOMATA package for calculating the finite state automata associated with a
short-lex automatic group. The aim is to provide an overview of the whole
process, rather than concentrating on technical details, which have been
already been published elsewhere. A number of related programs are also
described.
|
math
|
1,338 |
Exponential groups 2: Extensions of centralizers and tensor completion of CSA groups
|
math.GR
|
For a CSA group $G$ and a wide class of abelian groups $A$ we give an
explicit construction for the tensor $A$-completion of $G$ using free products
with amalgamations. We apply the obtained results to the study of basic
properties of $A$-free groups. In particular, canonical and reduced forms of
elements in $A$-free groups are introduced, and then commuting and conjugate
elements are described.
|
math
|
1,339 |
An alternative proof that the Fibonacci group F(2,9) is infinite
|
math.GR
|
This note contains a report of a proof by computer that the Fibonacci group
F(2,9) is automatic. The automatic structure can be used to solve the word
problem in the group. Furthermore, it can be seen directly from the
word-acceptor that the group generators have infinite order, which of course
implies that the group itself is infinite.
|
math
|
1,340 |
The chameleon groups of Richard J. Thompson: automorphisms and dynamics
|
math.GR
|
The automorphism groups of several of Thompson's countable groups of
piecewise linear homeomorphisms of the line and circle are computed and it is
shown that the outer automorphism groups of these groups are relatively small.
These results can be interpreted as stability results for certain structures of
PL functions on the circle. Machinery is developed to relate the structures on
the circle to corresponding structures on the line.
|
math
|
1,341 |
Disjunctive identities of finite groups and identities of regular representations
|
math.GR
|
In this paper we explicitly compute finite bases of disjunctive identities
and finite bases of regular representations for a number of interesting finite
groups.
|
math
|
1,342 |
Formal Languages and Infinite Groups
|
math.GR
|
This article is an introduction to formal languages from the point of view of
combinatorial group theory. Group theoretic applications are included and
language classes are defined algebraically.
|
math
|
1,343 |
A Shrinking Lemma for Indexed Languages
|
math.GR
|
This article presents a combinatorial result on indexed languages which was
inspired by an attempt to understand the structure of groups with indexed
language word problem. We show that a sufficiently long word in an indexed
language can be written as a product of a uniformly bounded number of terms in
such a way that some proper subproduct belongs to the language.
|
math
|
1,344 |
Generalized Small Cancellation Theory
|
math.GR
|
We present four generalized small cancellation conditions for finite
presentations and solve the word- and conjugacy problem in each case. Our
conditions $W$ and $W^*$ contain the non-metric small cancellation cases C(6),
C(4)T(4), C(3)T(6) (see [LS]) but are considerably more general. $W$ also
contains as a special case the small cancellation condition $W(6)$ of Juhasz
[J2]. If a finite presentation satisfies $W$ or $W^*$ then it has a quadratic
isoperimetric inequality and therefore solvable word problem. For the class $W$
this was first observed by Gersten in [G7] which also contains an idea of the
proof. Our main result here is the proof of the conjugacy problem for the
classes $W$ and $W^*$ which uses the geometry of non-positively curved
piecewise Euclidean complexes developed by Bridson in [Bri]. The conditions $V$
and $V^*$ generalize the small cancellation conditions C(7), C(5)T(4),
C(4)T(5), C(3)T(7). If a finite presentation satisfies the condition $V$ or
$V^*$, then it has a linear isoperimetric inequality and hence the group is
hyperbolic.
|
math
|
1,345 |
Commensurators of parabolic subgroups of Coxeter groups
|
math.GR
|
Let $(W,S)$ be a Coxeter system, and let $X$ be a subset of $S$. The subgroup
of $W$ generated by $X$ is denoted by $W_X$ and is called a parabolic subgroup.
We give the precise definition of the commensurator of a subgroup in a group.
In particular, the commensurator of $W_X$ in $W$ is the subgroup of $w$ in $W$
such that $wW_Xw^{-1}\cap W_X$ has finite index in both $W_X$ and $wW_Xw^{-1}$.
The subgroup $W_X$ can be decomposed in the form $W_X = W_{X^0} \cdot
W_{X^\infty} \simeq W_{X^0} \times W_{X^\infty}$ where $W_{X^0}$ is finite and
all the irreducible components of $W_{X^\infty}$" > are infinite. Let
$Y^\infty$ be the set of $t$ in $S$ such that $m_{s,t}=2$" > for all $s\in
X^\infty$. We prove that the commensurator of $W_X$ is $W_{Y^\infty} \cdot
W_{X^\infty} \simeq W_{Y^\infty} \times W_{X^\infty}$. In particular, the
commensurator of a parabolic subgroup is a parabolic subgroup, and $W_X$ is its
own commensurator if and only if $X^0=Y^\infty$.
|
math
|
1,346 |
Almost locally free groups and the genus question
|
math.GR
|
Sacerdote [Sa] has shown that the non-Abelian free groups satisfy precisely
the same universal-existential sentences Th(F$_2$)$\cap \forall \exists $ in a
first-order language L$_o$ appropriate for group theory. It is shown that in
every model of Th(F$_2$)$\cap \forall \exists $ the maximal Abelian subgroups
are elementarily equivalent to locally cyclic groups (necessarily nontrivial
and torsion free). Two classes of groups are interpolated between the
non-Abelian locally free groups and Remeslennikov's $\exists $-free groups.
These classes are the \textbf{almost locally free groups} and the
\textbf{quasi-locally free groups}. In particular, the almost locally free%
\textbf{\ }groups are the models of Th(F$_2$)$\cap \forall \exists $ while the
quasi-locally free groups are the $\exists $-free groups with maximal Abelian
subgroups elemenatarily equivalent to locally cyclic groups (necessarily
nontrivial and torsion free). Two principal open questions at opposite ends of
a spectrum are: (1.) Is every finitely generated almost locally free group
free? (2.) Is every quasi-locally free group almost locally free? Examples
abound of finitely generated quasi-locally free groups containing nontrivial
torsion in their Abelianizations. The question of whether or not almost locally
free groups have torsion free Abelianization is related to a bound in a free
group on the number of factors needed to express certain elements of the
derived group as a product of commutators.
|
math
|
1,347 |
Computing Nilpotent Quotients in Finitely Presented Lie Rings
|
math.GR
|
A nilpotent quotient algorithm for finitely presented Lie rings over Z
(LieNQ) is described. The paper studies graded and non-graded cases separately.
The algorithm computes the so-called nilpotent presentation for a finitely
presented, nilpotent Lie ring. The nilpotent presentation consists of
generators for the abelian group and the products---expressed as linear
combinations---for pairs formed by generators. Using that presentation the word
problem is decidable in $L$. Provided that the Lie ring $L$ is graded, it is
possible to determine the canonical presentation for a lower central factor of
$L$. LieNQ's complexity is studied and it is shown that optimizing the
presentation is NP-hard. Computational details are provided with examples,
timing and some structure theorems obtained from computations. Implementation
in C and GAP 3.5 interface is available.
|
math
|
1,348 |
Finitely presented subgroups of automatic groups and their isoperimetric functions
|
math.GR
|
We describe a general technique for embedding certain amalgamated products
into direct products. This technique provides us with a way of constructing a
host of finitely presented subgroups of automatic groups which are not even
asynchronously automatic. We can also arrange that such subgroups satisfy, at
best, an exponential isoperimetric inequality.
|
math
|
1,349 |
Infinite products of finite simple groups
|
math.GR
|
We classify those sequences $\langle S_{n} \mid n \in \mathbb{N} \rangle$ of
finite simple nonabelian groups such that the full product $\prod_{n} S_{n}$
has property (FA).
|
math
|
1,350 |
CSA groups and separated free constructions
|
math.GR
|
A group $G$ is said to be a {\it CSA}-group if all maximal abelian subgroups
of $G$ are malnormal. The class of CSA groups is of interest because it
contains torsion-free hyperbolic groups, groups acting freely on
$\Lambda$-trees and groups with the same existential theory as free groups. CSA
groups are also very closely related to the study of residually free groups and
tensor completions. In this paper we investigate which free constructions
(amalgamated products and HNN extensions) over CSA groups are again CSA. The
results are applied, in particular, to show that a torsion-free one-relator
group is CSA if and only if it does not contain nonabelian metabelin
Baumslag-Solitar groups and the direct product of the free group of rank 2 and
the infinite cyclic group.
|
math
|
1,351 |
Equations in a free Q-group
|
math.GR
|
In this work we investigate tensor completions of groups by associative
rings, which were introduced by R.Lyndon and G.Baumslag in 1960s. The main
result states that there exists an algorithm that decides if a given finite
system of equations over a free ${\bf Q}$-group has a solution, and if it does,
finds a solution. This statement can be generalized for ${\bf Q}$-completions
of torsion-free hyperbolic groups. Our proof significantly uses the techniques
of word hyperbolic groups and the results of E.Rips and Z.Sela on the
solvability of systems of equations in hyperbolic groups.
|
math
|
1,352 |
Hyperbolic groups and free constructions
|
math.GR
|
We investigate which free constructions (amalgamated products and
HNN-extensions) over word hyperbolic groups produce groups that are again word
hyperbolic. A complete answer is obtained for the case when the amalgamated
subgroups are virtually cyclic. The results are applied, in particular, to show
that a ${\bf Q}$-completion of a torsion-free hyperbolic group has solvable
word problem and conjugacy problem.
|
math
|
1,353 |
Fixed points of endomorphisms of a free metabelian group
|
math.GR
|
We consider IA-endomorphisms (i.e., Identical in Abelianization) of a free
metabelian group of finite rank, and give a matrix characterization of their
fixed points which is similar to (yet different from) the well-known
characterization of eigenvectors of a linear operator in a vector space. We
then use our matrix characterization to elaborate several properties of the
fixed point groups of metabelian endomorphisms. In particular, we show that the
rank of the fixed point group of an IA-endomorphism of the free metabelian
group of rank $n \ge 2$ can be either equal to 0, 1, or greater than $(n-1)$
(in particular, it can be infinite). We also point out a connection between
these properties of metabelian IA-endomorphisms and some properties of the
Gassner representation of pure braid groups.
|
math
|
1,354 |
Generalized primitive elements of a free group
|
math.GR
|
We study endomorphisms of a free group of finite rank by means of their
action on specific sets of elements. In particular, we prove that every
endomorphism of the free group of rank 2 which preserves an automorphic orbit
(i.e., acts ``like an automorphism" on one particular orbit), is itself an
automorphism. Then, we consider elements of a different nature, defined by
means of homological properties of the corresponding one-relator group. These
elements (``generalized primitive elements"), interesting in their own right,
can also be used for distinguishing automorphisms among arbitrary
endomorphisms.
|
math
|
1,355 |
Small cancellation groups and translation numbers
|
math.GR
|
In this paper we prove that C(4)-T(4)-P, C(3)-T(6)-P and C(6)-P small
cancellation groups are translation dis crete in the strongest possible sense
and that in these groups for any $g$ and any $n$ there is an algorithm deciding
whether or not the equation $ x^n=g$ has a solution. There is also an algorithm
for calculating for each $g$ the maximum $n$ such that $g$ is an $n$-th power
of some element. We also note that these groups cannot contain isomorphic
copies of the gr oup of $p$-adic fractions and so in particular of the group of
rational numbers. Besides we show that for C''(4)-T(4) and C''(3)-T(6) groups
all translation numbers are rational and have bounded denominators.
|
math
|
1,356 |
Some non-finitely presented Lie Algebras
|
math.GR
|
Let $L$ be a free Lie algebra over a field $k$, $I$ a non-trivial proper
ideal of $L$, $n>1$ an integer. The multiplicator $H_2(L/I^n,k)$ of $L/I^n$ is
not finitely generated, and so in particular, $L/I^n$ is not finitely
presented, even when $L/I$ is finite dimensional.
|
math
|
1,357 |
Rewriting Systems and Geometric 3-Manifolds
|
math.GR
|
The fundamental groups of most (conjecturally, all) closed 3-manifolds with
uniform geometries have finite complete rewriting systems. The fundamental
groups of a large class of amalgams of circle bundles also have finite complete
rewriting systems. The general case remains open.
|
math
|
1,358 |
Pascal's Triangles in Abelian and Hyperbolic Groups
|
math.GR
|
Pascal's triangle will give the number of geodesics from the identity to each
point of ${\bf Z}^2$ if you write it in each of the quadrants. Given a group
$G$ and generating set $\cal G$ we take the {\it Pascal's function} $p_{\cal
G}: G \to {\bf Z}_{\ge 0}$ to be the function which assigns to each $g\in G$
the number of geodesics from $1$ to $g$. We give a general method for
calculating this in hyperbolic groups and discuss the generic case in abelian
groups.
|
math
|
1,359 |
Some definition of the Artin exponent of finite groups
|
math.GR
|
The Artin exponent induced from cyclic subgroups of finite groups was studied
extensively by T.Y. Lam. A Burnside ring theoretic version of Lam's results for
$p$-groups was given by the author in an earlier paper. Here we look at the
Artin exponent induced from the elementary abelian subgroups of finite
$p$-groups using some results of A. Dress.
|
math
|
1,360 |
Class 2 Moufang loops, small Frattini Moufang loops, and code loops
|
math.GR
|
Let $L$ be a Moufang loop which is centrally nilpotent of class 2. We first
show that the nuclearly-derived subloop (normal associator subloop) $L^*$ of
$L$ has exponent dividing 6. It follows that $L_p$ (the subloop of $L$ of
elements of $p$-power order) is associative for $p>3$. Next, a loop $L$ is said
to be a {\it small Frattini Moufang loop}, or SFML, if $L$ has a central
subgroup $Z$ of order $p$ such that $C\isom L/Z$ is an elementary abelian
$p$-group. $C$ is thus given the structure of what we call a {\it coded vector
space}, or CVS. (In the associative/group case, CVS's are either orthogonal
spaces, for $p=2$, or symplectic spaces with attached linear forms, for $p>2$.)
Our principal result is that every CVS may be obtained from an SFML in this
way, and two SFML's are isomorphic in a manner preserving the central subgroup
$Z$ if and only if their CVS's are isomorphic up to scalar multiple.
Consequently, we obtain the fact that every SFM 2-loop is a code loop, in the
sense of Griess, and we also obtain a relatively explicit characterization of
isotopy in SFM 3-loops. (This characterization of isotopy is easily extended to
Moufang loops of class 2 and exponent 3.) Finally, we sketch a method for
constructing any finite Moufang loop which is centrally nilpotent of class 2.
|
math
|
1,361 |
Fractional Isoperimetric Inequalities and subgroup distortion
|
math.GR
|
It is shown that there exist infinitely many non-integers $r>2$ such that the
Dehn function of some finitely presented group is $\simeq n^r$. For each
positive rational number $s$ we construct pairs of finitely presented groups
$H\subset G$ such that the distortion of $H$ in $G$ is $\simeq n^s$. And for
each $s\ge 1$ we also construct finitely presented groups whose isodiametric
function is $\simeq n^s$.
|
math
|
1,362 |
Ping-Pong on Negatively Curved Groups
|
math.GR
|
We prove several generalisations of the ping-pong lemma for negatively curved
groups.
|
math
|
1,363 |
Solvable Baumslag-Solitar Groups Are Not Almost Convex
|
math.GR
|
The arguments of Cannon, Floyd, Grayson and Thurston showing that solve
geometry groups are not almost convex also show that solvable Baumslag-Solitar
groups are not almost convex.
|
math
|
1,364 |
Regular geodesic normal forms in virtually abelian groups
|
math.GR
|
Cannon has given an example of a virtually abelian group and a generating set
where the full language of geodesics is not regular. We describe a virtually
abelian group and a generating set so that no regular language of geodesics
surjects to the group.
|
math
|
1,365 |
Irreducible character degrees and normal subgroups
|
math.GR
|
Let N be a normal subgroup of a finite group G and consider the set cd(G|N)
of degrees of irreducible characters of G whose kernels do not contain N. A
number of theorems are proved relating the set cd(G|N) to the structure of N.
For example, if N is solvable, its derived length is bounded above by a
function of |cd(G|N)|. Also, if |cd(G|N)| is at most 2, then N is solvable and
its derived length is at most |cd(G|N)|. If G is solvable and |cd(G|N)| = 3,
then the derived length of N is at most 3.
|
math
|
1,366 |
Maximal subgroups of direct products
|
math.GR
|
We determine all maximal subgroups of the direct product $\sc G^n$ of $\sc n$
copies of a group~$\sc G$. If $\sc G$ is finite, we show that the number of
maximal subgroups of~$\sc G^n$ is a quadratic function of~$\sc n$ if $\sc G$ is
perfect, but grows exponentially otherwise. We~deduce a theorem of Wiegold
about the growth behaviour of the number of generators of~$\sc G^n$.
|
math
|
1,367 |
Duality and Local Group Cohomology
|
math.GR
|
Recently, Meierfrankenfeld has published three theorems on the cohomology of
a finitary module. They cover the local determination of complete reducibility;
the local splitting of group extensions; and the representation of locally
split extensions in the double dual. In this note we derive all three by
combining a certain duality between homology and cohomology with the continuity
of homology.
|
math
|
1,368 |
Amenability, Bilipschitz Maps, and the Von Neumann conjecture
|
math.GR
|
We determine when a quasi-isometry between discrete spaces is at bounded
distance from a bilipschitz map. From this we prove a geometric version of the
Von Neumann conjecture on amenability. We also get some examples in geometric
groups theory which show that the sign of the Euler characteristic is not a
coarse invariant. Finally we get some general results on uniformly finite
homology which we will apply to manifolds in a later paper.
|
math
|
1,369 |
A non-quasiconvexity embedding theorem for hyperbolic groups
|
math.GR
|
We show that if $G$ is a non-elementary torsion-free word hyperbolic group
then there exists another word hyperbolic group $G^*$, such that $G$ is a
subgroup of $G^*$ but $G$ is not quasiconvex in $G^*$.
|
math
|
1,370 |
Parallel poly pushdown groups
|
math.GR
|
We define a class of groups based on parallel computations by pushdown
automata. This class generalizes automatic groups. It includes the fundamental
groups of all 3-manifolds which obey Thurston' s geometrization conjecture. It
also includes nilpotent groups of arbitrary class and polynomial degree
isoperimetric inequality. It is closed under wreath product.
|
math
|
1,371 |
The ubiquity of Thompson's group F in groups of piecewise linear homeomorphisms of the unit interval
|
math.GR
|
We show that Thompson's group F occurs with great frequency in the group of
PL homeomorphisms of the unit interval.
|
math
|
1,372 |
Combinatorial methods: from groups to polynomial algebras
|
math.GR
|
Combinatorial methods (or methods of elementary transformations) came to
group theory from low-dimensional topology in the beginning of the century.
Soon after that, combinatorial group theory became an independent area with its
own powerful techniques. On the other hand, combinatorial commutative algebra
emerged in the sixties, after Buchberger introduced what is now known as
Gr\"{o}bner bases. The purpose of this survey is to show how ideas from one of
those areas contribute to the other.
|
math
|
1,373 |
A language theoretic analysis of combings
|
math.GR
|
A group is combable if it can be represented by a language of words
satisfying a fellow traveller property; an automatic group has a synchronous
combing which is a regular language. This paper gives a systematic analysis of
the properties of groups with combings in various formal language classes, and
of the closure properties of the associated classes of groups. It generalises
previous work, in particular of Epstein et al. and Bridson and Gilman.
|
math
|
1,374 |
Automatic groups associated with word orders other than shortlex
|
math.GR
|
The existing algorithm to compute and verify the automata associated with an
automatic group deals only with the subclass of shortlex automatic groups. This
paper describes the extension of the algorithm to deal with automatic groups
associated with other word orders (the algorithm has now been implemented ) and
reports on the use of the algorithm for specific examples; in particular a very
natural automatic (or asynchonously automatic) structure for the
Baumslag-Solitar and related classes of groups (closely related to one
described for some of those groups by Epstein et al. is found from a wreath
product order over shortlex.
|
math
|
1,375 |
On 2-generator subgroups of SO(3)
|
math.GR
|
We classify all subgroups of $SO(3)$ that are generated by two elements, each
a rotation of finite order, about axes separated by an angle that is a rational
multiple of $\pi$. In all cases we give a presentation of the subgroup. In most
cases the subgroup is the free product, or the amalgamated free product, of
cyclic groups or dihedral groups. The relations between the generators are all
simple consequences of standard facts about rotations by $\pi$ and $\pi/2$.
Embedded in the subgroups are explicit free groups on 2 generators, as used in
the Banach-Tarski paradox.
|
math
|
1,376 |
Automorphisms of generalized Thompson groups
|
math.GR
|
We look at the automorphisms of Thompson type groups of piecewise linear
homeomorphisms of the real line or circle that use slopes that are integral
powers of a fixed integer n with n>2. We show that large numbers of "exotic"
automorphisms appear---automorphisms that are represented as conjugation by
non-PL homeomorphisms of the real line or circle. This is in contrast to the
n=2 case where no such automorphisms appear.
|
math
|
1,377 |
An Endomorphism of a Finitely Generated Residually Finite Group
|
math.GR
|
Let $\phi:G\rightarrow G$ be an endomorphism of a finitely generated
residually finite group. R.~Hirshon asked if there exists~$n$ such that the
restriction of $\phi$ to $\phi^n(G)$ is injective. We give an example to show
that this is not always the case.
|
math
|
1,378 |
Doubles of groups and hyperbolic LERF 3-manifolds
|
math.GR
|
We show that the quasiconvex subgroups in doubles of certain negatively
curved groups are closed in the profinite topology. This allows us to construct
the first known large family of hyperbolic 3-manifolds such that any finitely
generated subgroup of the fundamental group of any member of the family is
closed in the profinite topology.
|
math
|
1,379 |
Combinatorial problems about free groups and algebras
|
math.GR
|
This is a survey of recent progress in several areas of combinatorial
algebra. We consider combinatorial problems about free groups, polynomial
algebras, free associative and Lie algebras. Our main idea is to study
automorphisms and, more generally, homomorphisms of various algebraic systems
by means of their action on ``very small" sets of elements, as opposed to a
traditional approach of studying their action on subsystems (like subgroups,
normal subgroups; subalgebras, ideals, etc.) We will show that there is a lot
that can be said about a homomorphism, given its action on just a single
element, if this element is ``good enough". Then, we consider somewhat bigger
sets of elements, like, for example, automorphic orbits, and study a variety of
interesting problems arising in that framework.
One more point that we make here is that one can use similar combinatorial
ideas in seemingly distant areas of algebra, like, for example, group theory
and commutative algebra. In particular, we use the same language of
``elementary transformations" in different contexts and show that this approach
appears to be quite fruitful for all the areas involved.
|
math
|
1,380 |
The automorphism tower of a free group
|
math.GR
|
We prove that the automorphism group of an arbitrary non-abelian free group
is complete. It generalizes the result by J.Dyer and E.Formanek (1975) stating
the completeness of automorphism group of finitely generated free groups. Using
the description of involutions in automorphism groups of free groups (J. Dyer,
P. Scott, 1975) we obtain a group-theoretic characterization of inner
automorphisms determined by primitive elements in the automorphism group of any
non-abelian free group F. It follows that the subgroup Inn(F) is characteristic
in Aut(F), and hence the latter one is complete.
|
math
|
1,381 |
Set theory is interpretable in the automorphism group of a free group
|
math.GR
|
In 1976 S. Shelah posed the following problem: for which variety V of
algebras the automorphism group of any free algebra F from V of "large"
infinite rank interprets by means of first-order logic set theory (according to
his results, for every variety V the endomorphism semi-group of F interprets
set theory if rank(F) is an infinite cardinal greater than the power of the
language of V). There are examples of varieties for which the answer is
negative; one such an example, the variety of all algebras in empty language,
is due to Shelah (1973). The author earlier showed that the answer is positive
for any variety of vector spaces over a fixed division ring. In the present
paper it is proved that the same holds for the variety of all groups: the
automorphism group of any infinitely generated free group F interprets set
theory. It follows, in particular, that the group Aut(F) is as undecidable as
possible.
|
math
|
1,382 |
Bilinear maps and central extensions of abelian groups
|
math.GR
|
We show that every nilpotent group of class at most two may be embedded in a
central extension of abelian groups with bilinear cocycle. The embedding is
shown to depend only on the base group. Some refinements are obtained by
considering the cohomological situation explicitly.
|
math
|
1,383 |
Automorphisms of one-relator groups
|
math.GR
|
It is a well-known fact that every group $G$ has a presentation of the form
$G = F/R$, where $F$ is a free group and $R$ the kernel of the natural
epimorphism from $F$ onto $G$. Driven by the desire to obtain a similar
presentation of the group of automorphisms $Aut(G)$, we can consider the
subgroup $Stab(R) \subseteq Aut(F)$ of those automorphisms of $F$ that
stabilize $R$, and try to figure out if the natural homomorphism $Stab(R) \to
Aut(G)$ is onto, and if it is, to determine its kernel. Both parts of this task
are usually quite hard. The former part received considerable attention in the
past, whereas the latter, more difficult, part (determining the kernel) seemed
unapproachable. Here we approach this problem for a class of one-relator groups
with a special kind of small cancellation condition. Then, we address a
somewhat easier case of 2-generator (not necessarily one-relator) groups, and
determine the kernel of the above mentioned homomorphism for a rather general
class of those groups.
|
math
|
1,384 |
Quasi-isometrically embedded subgroups of Thompson's group F
|
math.GR
|
The goal of this paper is to construct quasi-isometrically embedded subgroups
of Thompson's group $F$ which are isomorphic to $\fz^n$ for all $n$. A result
estimating the norm of an element of Thompson's group is found. As a corollary,
Thompson's group is seen to be an example of a finitely presented group which
has an infinite-dimensional asymptotic cone.
|
math
|
1,385 |
Class Operators as Intertwining Maps into the Group Algebra
|
math.GR
|
With the aim of completing the previous study by A. Or{\l}owski and the
author concerning intertwining maps between induced representations and
conjugation representation, termed here weighted class operators, we compute
the latter explicitely for the conjugation representation arising from the
regular representation in the group algebra of a compact group. To that efect a
theorem of Wigner-- Eckart type for weighted class operators obtained from
matrix coefficients of irreducible representations of a compact group is
proved. Also the previous construction of weighted class operators is reviewed
and extended to the case of locally compact groups rather then just compact
ones. Submitted for: Proceedings of the II International Workshop "Lie Theory
and Its Applications in Physics", August 1997, Clausthal.
|
math
|
1,386 |
Absolutely closed nil-2 groups
|
math.GR
|
Using the description of dominions in the variety of nilpotent groups of
class at most two, we give a characterization of which groups are absolutely
closed in this variety. We use the general result to derive an easier
characterization for some subclasses; e.g. an abelian group $G$ is absolutely
closed in ${\cal N}_2$ if and only if $G/pG$ is cyclic for every prime $p$.
|
math
|
1,387 |
Liftez les Sylows! Une suite à ``Sous-groupes periodiques d'un groupe stable''
|
math.GR
|
If $G$ is an omega-stable group with a normal definable subgroup $H$, then
the Sylow-$2$-subgroups of $G/H$ are the images of the Sylow-$2$-subgroups of
$G$.
|
math
|
1,388 |
Dominions in varieties of nilpotent groups
|
math.GR
|
We investigate the concept of dominion (in the sense of Isbell) in several
varieties of nilpotent groups. We obtain a full description of dominions in the
variety of nilpotent groups of class at most two. Then we look at the behavior
of dominions of subgroups of groups in ${\cal N}_2$ when taken in the context
of ${\cal N}_c$ with $c>2$. Finally we establish the existence of nontrivial
dominions in the category of all nilpotent groups.
|
math
|
1,389 |
Automatic Groups and Knuth-Bendix with Infinitely Many Rules
|
math.GR
|
It is shown how to use a small finite state automaton in two variables in
order to carry out part of the Knuth--Bendix process for rewriting words in a
group. The main objective is to provide a substitute for the most
space-demanding module of the existing software which attempts to find a
shortlex-automatic structure for a group. The two-variable automaton can be
used to store an infinite set of rules and to carry out fast reduction of
arbitrary words using this infinite set. We introduce a new operation, which we
call welding, which applies to an arbitrary finite state automaton. In our
context this operation is vital. We point out a small potential improvement in
the subset algorithm for making a non-deterministic automaton deterministic.
|
math
|
1,390 |
Dominions in finitely generated nilpotent groups
|
math.GR
|
In the first part, we prove that the dominion (in the sense of Isbell) of a
subgroup of a finitely generated nilpotent group is trivial in the category of
all nilpotent groups. In the second part, we show that the dominion of a
subgroup of a finitely generated nilpotent group of class two is trivial in the
category of all metabelian nilpotent groups.
|
math
|
1,391 |
A generalized argument for dominions in varieties of groups
|
math.GR
|
An argument used to show that certain varieties of nilpotent groups have
instances of nontrivial dominions is considered, and generalized. The same is
done with the argument used to show that there are nontrivial dominions in the
variety of metabelian groups, to suggest how this general technique may be
used.
|
math
|
1,392 |
Dominions in the variety of metabelian groups
|
math.GR
|
This paper has been withdrawn. The results are now part of math.GR/9804072.
|
math
|
1,393 |
Nonsurjective epimorphisms in decomposable varieties of groups
|
math.GR
|
A full characterization of when a subgroup $H$ of a group $G$ in a varietal
product ${\cal NQ}$ is epimorphically embedded in $G$ (in the variety ${\cal
NQ}$) is given. From this, a result of S.~McKay is derived, which states that
if ${\cal NQ}$ has instances of nonsurjective epimorphisms, then ${\cal N}$
also has instances of nonsurjective epimorphisms. Two partial converses to
McKay's result are also given: when~$G$ is a finite nonabelian simple group;
and when~$G$ is finite and ${\cal Q}$ is a product of varieties of nilpotent
groups, each of which contains the infinite cyclic group.
|
math
|
1,394 |
Dominions in decomposable varieties
|
math.GR
|
Dominions, in the sense of Isbell, are investigated in the context of
decomposable varieties of groups. An upper and lower bound for dominions in
such a variety is given in terms of the two varietal factors, and the internal
structure of the group being analyzed. Finally, the following result is
established: If a variety ${\cal N}$ has instances of nontrivial dominions,
then for any proper subvariety ${\cal Q}$ of ${\cal G}roup$, ${\cal NQ}$ also
has instances of nontrivial dominions.
|
math
|
1,395 |
Dominions in varieties generated by simple groups
|
math.GR
|
Let~$S$ be a finite nonabelian simple group, and let $H$ be a subgroup of
$S$. In this work, the dominion (in the sense of Isbell) of $H$ in $S$ in
rmVar(S)$ is determined, generalizing an example of B.H. Neumann. A necessary
and sufficient condition for $H$ to be epimorphically embedded in $S$ is
obtained. These results are then extended to a variety generated by a family of
finite nonabelian simple groups.
|
math
|
1,396 |
A Bound for the Nilpotency Class of a Finite p-Group in terms of its Coexponent
|
math.GR
|
The coexponent of a finite p-group is introduced and we consider how the
nilpotency class is bounded in terms of this invariant.
|
math
|
1,397 |
Boundaries of strongly accessible hyperbolic groups
|
math.GR
|
We consider splittings of groups over finite and two-ended subgroups. We
study the combinatorics of such splittings using generalisations of Whitehead
graphs. In the case of hyperbolic groups, we relate this to the topology of the
boundary. In particular, we give a proof that the boundary of a one-ended
strongly accessible hyperbolic group has no global cut point.
|
math
|
1,398 |
Automatic groups, subgroups and cosets
|
math.GR
|
The history, definition and principal properties of automatic groups and
their generalisations to subgroups and cosets are reviewed briefly, mainly from
a computational perspective. A result about the asynchronous automaticity of an
HNN extension is then proved and applied to an example that was proposed by
Mark Sapir.
|
math
|
1,399 |
Hairdressing in groups: a survey of combings and formal languages
|
math.GR
|
A group is combable if it can be represented by a language of words
satisfying a fellow traveller property; an automatic group has a synchronous
combing which is a regular language. This article surveys results for combable
groups, in particular in the case where the combing is a formal language.
|
math
|
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