2,099 research outputs found
Proving finitely presented groups are large by computer
We present a theoretical algorithm which, given any finite presentation of a
group as input, will terminate with answer yes if and only if the group is
large. We then implement a practical version of this algorithm using Magma and
apply it to a range of presentations. Our main focus is on 2-generator
1-relator presentations where we have a complete picture of largeness if the
relator has exponent sum zero in one generator and word length at most 12, as
well as when the relator is in the commutator subgroup and has word length at
most 18. Indeed all but a tiny number of presentations define large groups.
Finally we look at fundamental groups of closed hyperbolic 3-manifolds, where
the algorithm readily determines that a quarter of the groups in the Snappea
closed census are large.Comment: 37 pages including 6 pages of table
Strictly ascending HNN extensions of finite rank free groups that are linear over Z
We find strictly ascending HNN extensions of finite rank free groups
possessing a presentation 2-complex which is a non positively curved square
complex. On showing these groups are word hyperbolic, we have by results of
Wise and Agol that they are linear over the integers. An example is the
endomorphism of the free group on a,b with inverses A,B that sends a to aBaab
and b to bAbba.Comment: 21 pages, just 1 figur
Non proper HNN extensions and uniform uniform exponential growth
If a finitely generated torsion free group K has the property that all
finitely generated subgroups S of K are either small or have growth constant
bounded uniformly away from 1 then a non proper HNN extension G of K, that is a
semidirect product of K by the integers, has the same property. Here small
means cyclic or, if the automorphism has no periodic conjugacy classes, free
abelian of bounded rank.Comment: 29 page
Strictly ascending HNN extensions in soluble groups
We show that there exist finitely generated soluble groups which are not LERF
but which do not contain strictly ascending HNN extensions of a cyclic group.
This solves Problem 16.2 in the Kourovka notebook. We further show that there
is a finitely presented soluble group which is not LERF but which does not
contain a strictly ascending HNN extension of a polycyclic group.Comment: 10 page
Balanced groups and graphs of groups with infinite cyclic edge groups
We give a necessary and sufficient condition for the fundamental group of a
finite graph of groups with infinite cyclic edge groups to be acylindrically
hyperbolic, from which it follows that a finitely generated group splitting
over Z cannot be simple. We also give a necessary and sufficient condition
(when the vertex groups are torsion free) for the fundamental group to be
balanced, where a group is said to be balanced if conjugate to
implies that for all infinite order elements
Acylindrical hyperbolicity, non simplicity and SQ-universality of groups splitting over Z
We show, using acylindrical hyperbolicity, that a finitely generated group
splitting over cannot be simple. We also obtain SQ-universality in most
cases, for instance a balanced group (one where if two powers of an infinite
order element are conjugate then they are equal or inverse) which is finitely
generated and splits over must either be SQ-universal or it is one of
exactly seven virtually abelian exceptions.Comment: Much shorter version of 1509.05688 with strengthening of main resul
Groups possessing only indiscrete embeddings in SL(2,C)
We give results on when a finitely generated group has only indiscrete
embeddings in SL(2,C), with particular reference to 3-manifold groups. For
instance if we glue two copies of the figure 8 knot along its torus boundary
then the fundamental group of the resulting closed 3-manifold sometimes embeds
in SL(2,C) and sometimes does not, depending on the identification. We also
give another quick counterexample to Minsky's simple loop question.Comment: Minor changes and update
Virtual finite quotients of finitely generated groups
If G is a semidirect product N by H with N normal and finitely generated then
G has the property that every finite group is a quotient of some finite index
subgroup of G if and only if one of N and H has this property. This has
applications to 3-manifolds and to cyclically presented groups, for instance
for any fibred hyperbolic 3-manifold M and any finite simple group S, there is
a cyclic cover of M whose fundamental group surjects to S. We also give a short
proof of the residual finiteness of ascending HNN extensions of finite rank
free groups when the induced map on homology is injective
Fibred and Virtually Fibred hyperbolic 3-manifolds in the censuses
Following on from work of Dunfield, we determine the fibred status of all the
unknown hyperbolic 3-manifolds in the cusped census. We then find all the
fibred hyperbolic 3-manifolds in the closed census and use this to find over
100 examples each of closed and cusped virtually fibred non-fibred census
3-manifolds, including the Weeks manifold. We also show that the co-rank of the
fundamental group of every 3-manifold in the cusped and in the closed census is
0 or 1.Comment: 50 pages, including 2 figures and 14 pages of table
Large mapping tori of free group endomorphisms
We present an algorithm which, given any finite presentation of a group as
input, will terminate with answer yes if and only if the group is large. We use
this to prove that a mapping torus of a finitely generated free group
automorphism is large if it contains the integers times the integers as a
subgroup of infinite index. We then extend this result to mapping tori of
finitely generated free group endomorphisms, as well as showing that such a
group is large if it contains a Baumslag-Solitar group of infinite index and
has a finite index subgroup with first Betti number at least 2. We also show
that if a group possesses a deficiency 1 presentation where one of the relators
is a commutator then it is the integers times the integers, or it is large, or
it is as far as possible from being residually finite.Comment: 41 pages with no figure
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