225 research outputs found
Automatic groups and amalgams
AbstractThe objectives of this paper are twofold. The first is to provide a self-contained introduction to the theory of automatic and asynchronously automatic groups, which were invented a few years ago by J.W. Cannon, D.B.A. Epstein, D.F. Holt, M.S. Paterson and W.P. Thurston. The second objective is to prove a number of new results about the construction of new automatic and asynchronously automatic groups from old ones by means of amalgamated products
Finitely presented wreath products and double coset decompositions
We characterize which permutational wreath products W^(X)\rtimes G are
finitely presented. This occurs if and only if G and W are finitely presented,
G acts on X with finitely generated stabilizers, and with finitely many orbits
on the cartesian square X^2. On the one hand, this extends a result of G.
Baumslag about standard wreath products; on the other hand, this provides
nontrivial examples of finitely presented groups. For instance, we obtain two
quasi-isometric finitely presented groups, one of which is torsion-free and the
other has an infinite torsion subgroup.
Motivated by the characterization above, we discuss the following question:
which finitely generated groups can have a finitely generated subgroup with
finitely many double cosets? The discussion involves properties related to the
structure of maximal subgroups, and to the profinite topology.Comment: 21 pages; no figure. To appear in Geom. Dedicat
Automatic structures, rational growth and geometrically finite hyperbolic groups
We show that the set of equivalence classes of synchronously
automatic structures on a geometrically finite hyperbolic group is dense in
the product of the sets over all maximal parabolic subgroups . The
set of equivalence classes of biautomatic structures on is
isomorphic to the product of the sets over the cusps (conjugacy
classes of maximal parabolic subgroups) of . Each maximal parabolic is a
virtually abelian group, so and 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 is
regular. Moreover, the growth function of 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.Comment: Plain Tex, 26 pages, no figure
Discriminating Groups
A group G is termed discriminating if every group separated by G is discriminated by G. In this paper we answer several questions concerning discrimination which arose from [2]. We prove that a finitely generated equationally Noetherian group G is discriminating if and only if the quasivariety generated by G is the minimal universal class containing G. Among other results, we show that the non-abelian free nilpotent groups are non-discriminating. Finally we list some open problems concerning discriminating groups
Algebraic Geometry over Free Metabelian Lie Algebra I: U-Algebras and Universal Classes
This paper is the first in a series of three, the aim of which is to lay the
foundations of algebraic geometry over the free metabelian Lie algebra . In
the current paper we introduce the notion of a metabelian Lie -algebra and
establish connections between metabelian Lie -algebras and special matrix
Lie algebras. We define the -localisation of a metabelian Lie
-algebra and the direct module extension of the Fitting's radical of
and show that these algebras lie in the universal closure of .Comment: 34 page
Splittings of generalized Baumslag-Solitar groups
We study the structure of generalized Baumslag-Solitar groups from the point
of view of their (usually non-unique) splittings as fundamental groups of
graphs of infinite cyclic groups. We find and characterize certain
decompositions of smallest complexity (`fully reduced' decompositions) and give
a simplified proof of the existence of deformations. We also prove a finiteness
theorem and solve the isomorphism problem for generalized Baumslag-Solitar
groups with no non-trivial integral moduli.Comment: 20 pages; hyperlinked latex. Version 2: minor change
Dimension of the Torelli group for Out(F_n)
Let T_n be the kernel of the natural map from Out(F_n) to GL(n,Z). We use
combinatorial Morse theory to prove that T_n has an Eilenberg-MacLane space
which is (2n-4)-dimensional and that H_{2n-4}(T_n,Z) is not finitely generated
(n at least 3). In particular, this recovers the result of Krstic-McCool that
T_3 is not finitely presented. We also give a new proof of the fact, due to
Magnus, that T_n is finitely generated.Comment: 27 pages, 9 figure
Conjugacy in Baumslag's group, generic case complexity, and division in power circuits
The conjugacy problem belongs to algorithmic group theory. It is the
following question: given two words x, y over generators of a fixed group G,
decide whether x and y are conjugated, i.e., whether there exists some z such
that zxz^{-1} = y in G. The conjugacy problem is more difficult than the word
problem, in general. We investigate the complexity of the conjugacy problem for
two prominent groups: the Baumslag-Solitar group BS(1,2) and the
Baumslag(-Gersten) group G(1,2). The conjugacy problem in BS(1,2) is
TC^0-complete. To the best of our knowledge BS(1,2) is the first natural
infinite non-commutative group where such a precise and low complexity is
shown. The Baumslag group G(1,2) is an HNN-extension of BS(1,2). We show that
the conjugacy problem is decidable (which has been known before); but our
results go far beyond decidability. In particular, we are able to show that
conjugacy in G(1,2) can be solved in polynomial time in a strongly generic
setting. This means that essentially for all inputs conjugacy in G(1,2) can be
decided efficiently. In contrast, we show that under a plausible assumption the
average case complexity of the same problem is non-elementary. Moreover, we
provide a lower bound for the conjugacy problem in G(1,2) by reducing the
division problem in power circuits to the conjugacy problem in G(1,2). The
complexity of the division problem in power circuits is an open and interesting
problem in integer arithmetic.Comment: Section 5 added: We show that an HNN extension G = < H, b | bab^-1 =
{\phi}(a), a \in A > has a non-amenable Schreier graph with respect to the
base group H if and only if A \neq H \neq
L^2-Betti numbers of one-relator groups
We determine the L^2-Betti numbers of all one-relator groups and all
surface-plus-one-relation groups (surface-plus-one-relation groups were
introduced by Hempel who called them one-relator surface groups). In particular
we show that for all such groups G, the L^2-Betti numbers b_n^{(2)}(G) are 0
for all n>1. We also obtain some information about the L^2-cohomology of
left-orderable groups, and deduce the non-L^2 result that, in any
left-orderable group of homological dimension one, all two-generator subgroups
are free.Comment: 18 pages, version 3, minor changes. To appear in Math. An
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