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Group Theory / Théorie des groupes
(Olga Kharlampovich, Organizer)
- ROBERT BURNS, York
Constructible groups and a dichotomy in group laws
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The class of ``constructible'' groups is obtained by closing the class
of finite and soluble groups under the operations of forming
extensions, subgroups, Cartesian products, and the ``local'' operator,
i.e. inclusion of all groups which are locally among those
already obtained. We characterize (almost) those group laws w = 1 such
that every constructible group satisfying that law is
nilpotent-by-locally finite of finite exponent. Thus semigroup laws and
Engel laws represent special cases of such laws.
- LISA CARBONE, Harvard University
Lattices on non-uniform trees
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Let x be a locally finite tree. Then G = Aut(X) is a locally compact
group. An `X-lattice' is a discrete subgroup L of G such that
L\X has `finite volume'. We call L a cocompact lattice if
L\X is finite, and a non-uniform lattice otherwise. If G
is unimodular and G\X is finite, then Bass and Kulkarni
([BK]) have shown that G contains a cocompact X-lattice. It has
been shown by the author ([C1]) that if G contains a cocompact
lattice, is not discrete and acts minimally on X, then G also
contains a non-uniform X-lattice. If we drop the assumption that
G\X is finite, and assume only that G is unimodular, and
that G\X is infinite but has `finite volume', then Bass and
Lubotzky conjecture ([BL]) that G still contains a non-uniform
X-lattice. We provide substantial evidence that this conjecture is
true by proving it in a large number of cases ([C2]).
References
[BK ] H. Bass and R. Kulkarni, Uniform tree lattices.
J. Amer. Math. Soc. (4) 3(1990).
[BL ] H. Bass A. Lubotzky, Tree lattices.
Birkhauser, 2000, to appear.
[C1 ] L. Carbone, Non-uniform lattices on uniform trees.
Mem. Amer. Math. Soc., 2000, to appear.
[C2 ] , Lattices on non-uniform trees.
2000, in preparation.
- JOHN DIXON, School of Mathematics and Statistics, Carleton University,
Ottawa, Ontario K1S 5B6
Using probabilistic methods to prove theorems in group theory
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Since the 1960s there has been a series of results which state that
certain events hold ``almost always'' in a group (or in a sequence of
finite groups as the order tends to infinity). An early result of this
type (conjectured by E. Netto around 1900), was proved by the speaker
in 1969 [1]: If two elements x, y are selected uniformly at random
from the symmetric group Sn, then the probability that the
alternating group An is contained in the subgroup áx,yñ tends to 1 as n® ¥. The object of this
talk is to survey some of these results and to explain how they have
been used recently to prove theorems in infinite groups which appear to
have no ``probabilistic'' component. For example, it is shown in [2]
how results of this type can be used to give an alternative proof of a
theorem of T. S. Weigel [3-5]: A free group of rank n ³ 2 is
residually S for any infinite set S of finite
simple groups.
References
J. D. Dixon, The probability of generating the symmetric group.
Math. Z. 110(1969), 199-205.
J. D. Dixon, L. Pyber, Á. Seress and S. Shalev, Residual
properties of free groups and probabilistic methods. submitted for
publication.
T. S. Weigel, Residual properties of free groups. J. Algebra
160(1993), 16-41.
, Residual properties of free groups II. Comm.
Algebra 20(1992), 1395-1425.
, Residual properties of free groups III.
Israel J. Math. 77(1992), 65-81.
- ANTHONY M. GAGLIONE, US Naval Academy
Are some groups less discriminating than others
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Here we show that the free nilpotent group of class c and rank r is
not discriminating if min{c,r} is at least 2. We give the
necessary preliminaries in order to understand the result.
- STEVE GERSTEN, Utah, USA
To be announced
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- BOB GILMAN
Languages of free generators
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It is well known that every subgroup of a finitely generated free group
admits a set of Nielsen-reduced free generators with central significant
factors. However, the situation changes if we require the set of
generators to lie in a prescribed class of formal languages. We study the
effect of this requirement and apply our results to a characterization of
automatic groups.
- SERGEI IVANOV, University of Illinois at Urbana-Champaign, Urbana,
Illinois 61821, USA
Intersecting free subgroups in free products of groups
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A subgroup H of a free product Õa Î I* Ga of
groups Ga, a Î I, is called factor free if for
every S Î Õa Î I* Ga and b Î I one
has S H S-1 ÇGb = { 1} (by Kurosh's theorem on
subgroups of free products, factor free subgroups are free). If K
is a finitely generated free group, denote [`(r)](K) = max(r(K)-1,0), where r(K) is the rank of K.
It is proved that if H, K are finitely generated factor free
subgroups of a free product Õa Î I* Ga then
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and this estimate is sharp and cannot be improved. On the other hand,
it is shown that if the factors Ga, a Î I, have no
nontrivial elements of order less than or equal to q, where q £ 6, and H, K are finitely generated factor free subgroups of the
free product Õa Î I* Ga then
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(H ÇK) £ 2 |
q+1
q-1
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In particular, if the factors Ga, a Î I, are torsion
free groups then [`(r)](H ÇK) £ 2.8 [`(r)](H) [`(r)](K). It is
also proved that if the factors Ga, a Î I, are
linearly ordered groups then, as for free groups, one has
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- OLGA KHARLAMPOVICH, McGill University
JSJ decomposition for finitely generated fully residually free
groups
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We will construct an algorithm to obtain the canonical JSJ
decomposition for freely indecomposable finitely generated fully
residually free groups. We also construct canonical embeddings of such
groups into NTQ (nondegenerate triangular quasiquadratic) groups.
- JOHN LABUTE, McGill University
On the cohomological dimension of one-relator pro-p-groups
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Abstract: One of the central outstanding problems for pro-p-groups with
one relator is the determination of their cohomological dimension. In
particular, is such a group of cohomological dimension 1 or 2 if
the group is torsion free? In our talk we will give a survey of known
results concerning this problem.
- ALEXEI MIASNIKOV, City College of CUNY
Exploiting links between the algebraic geometry over groups and
logic
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It turns out that the basic notions of the algebraic geometry over
groups have some natural counterparts in the model theory and the
universal algebra. On the one hand, this brings powerful methods of the
model theory into the algebraic geometry and shows how to develop this
type of geometry in a more general framework. On the other hand, these
connections put new perspectives for some known methods in the
universal algebra and lead to several new ideas in this area.
As an example, I will discuss new interesting links between the old
Malcev's problem from the model theory and two recent problems in
algebraic geometry which are due to B. Plotkin.
These are joint results with Vladimir Remesslennikov.
- AKBAR RHEMTULLA, Alberta
Incompatible group orderings, local indicability and braids
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Following a brief survey of results on which right orderable (RO)
groups are locally indicable (LI), we go on to discuss which right
orders on a subgroup H of of finite index in a RO-group G can
be extended to right orders of G when G is not a LI-group. The
results are applied to Artin Braid groups Bn, n ³ 5, which are
known to be RO but not LI and the pure braids Pn which are of
finite index in Bn and known to be twisted orderable.
- LUIS RIBES, Carleton
Finite extensions of free pro-p groups
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A description of virtually free pro-p groups having a free pro-p
subgroup of index p is given in terms of free products. This extends
a well-known result of Serre (torsion-free virtually free pro-p
groups are free pro-p). It has also some consequences on the
description of the fixed points of automorphisms of p-power order of
free pro-p groups.
- VLADIMIR SHPILRAIN, City College of CUNY
Measuring sets in finitely generated groups
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To measure a finite (sub)set, one just counts elements of this
(sub)set. To measure an infinite (sub)set, one can use different
methods; one method that is becoming increasingly popular is
considering intersections of the given infinite (sub)set with finite
sets of ``uniformly'' increasing size, and then taking the limit of the
size of such intersections. We use this method to measure various
interesting subsets of the set of all elements of a given finitely
generated groups. Based on these measurements, we address various
probabilistic questions, like, for example: what is the probability
for a randomly chosen element of a given group to be a generator? To
belong to the commutator subgroup?, etc.
Answers to some of those questions turn out to be non-trivial, and
sometimes even unexpected.
- FRANCIS CHI-YU TANG, University of Waterloo
On the residual finiteness of the outer automorphism groups of
certain generalized free products
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It is a well-known result of G. Baumslag that automorphism groups of
finitely generated residually finite groups are residually finite. Not
much is known about the residual finiteness of outer automorphism
groups of finitely generated residually finite groups. E. Grossman
(1974) showed that the outer automorphism groups of surface groups are
residually finite. Using recent results on conjugacy separability, we
showed that the outer automorphisms of certain Seifert manifolds and
generalized free products of free groups with cyclic amalgamation are
residually finite.
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