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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 ... | 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 pr... | 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 previousl... | 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 ... | 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 clo... | 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 grou... | 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 explici... | 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 thi... | 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 increasi... | 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 resolut... | 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 th... | 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 ... | 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 pro... | 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... | 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.... | 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 ... | 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 s... | 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 computa... | 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 ... | 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 s... | 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 $M_{22}$, $M_{23}$, $M_{24}$,
$He$, $Co_1$, $Co_2$,... | 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 ... | 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... | 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
cyclomat... | 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 crosse... | 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
occurren... | 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 know... | 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 pr... | 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 ... | 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 publish... | 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 g... | 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 i... | 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 o... | 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 ... | 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 ca... | 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 ... | 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 elemen... | 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 abeli... | 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 expo... | 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 close... | 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.... | 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 ${\bf Q}$-co... | 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 cha... | 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 ... | 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 calculat... | 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 ... | 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 usin... | 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 ... | 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 $s\ge 1$ w... | 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|... | 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 g... | 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 cer... | 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 invaria... | 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 inequ... | 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 th... | 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 pr... | 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
report... | 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... | 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 homeomorphi... | 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 p... | 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... | 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. Sco... | 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)... | 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 ... | 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... | 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 i... | 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 ${\cal N}_2$... | 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 ${\ca... | 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 ... | 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 ... | 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 inst... | 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 varie... | 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 resul... | 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 access... | 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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