75,976 research outputs found
On the Finiteness Property for Rational Matrices
We analyze the periodicity of optimal long products of matrices. A set of
matrices is said to have the finiteness property if the maximal rate of growth
of long products of matrices taken from the set can be obtained by a periodic
product. It was conjectured a decade ago that all finite sets of real matrices
have the finiteness property. This conjecture, known as the ``finiteness
conjecture", is now known to be false but no explicit counterexample to the
conjecture is available and in particular it is unclear if a counterexample is
possible whose matrices have rational or binary entries. In this paper, we
prove that finite sets of nonnegative rational matrices have the finiteness
property if and only if \emph{pairs} of \emph{binary} matrices do. We also show
that all {pairs} of binary matrices have the finiteness property.
These results have direct implications for the stability problem for sets of
matrices. Stability is algorithmically decidable for sets of matrices that have
the finiteness property and so it follows from our results that if all pairs of
binary matrices have the finiteness property then stability is decidable for
sets of nonnegative rational matrices. This would be in sharp contrast with the
fact that the related problem of boundedness is known to be undecidable for
sets of nonnegative rational matrices.Comment: 12 pages, 1 figur
Subdirect products of groups and the n-(n+1)-(n+2) Conjecture
We analyse the subgroup structure of direct products of groups. Earlier work
on this topic has revealed that higher finiteness properties play a crucial
role in determining which groups appear as subgroups of direct products of free
groups or limit groups. Here, we seek to relate the finiteness properties of a
subgroup to the way it is embedded in the ambient product. To this end we
formulate a conjecture on finiteness properties of fibre products of groups. We
present different approaches to this conjecture, proving a general result on
finite generation of homology groups of fibre products and, for certain special
cases, results on the stronger finiteness properties F_n and FP_n.Comment: 32 page
Abels's groups revisited
We generalize a class of groups introduced by Herbert Abels to produce
examples of virtually torsion free groups that have Bredon-finiteness length
m-1 and classical finiteness length n-1 for all 0 < m <= n.
The proof illustrates how Bredon-finiteness properties can be verified using
geometric methods and a version of Brown's criterion due to Martin Fluch and
the author.Comment: 17 pages, 2 figures, v2 more detaile
An explicit counterexample to the Lagarias-Wang finiteness conjecture
The joint spectral radius of a finite set of real matrices is
defined to be the maximum possible exponential rate of growth of long products
of matrices drawn from that set. A set of matrices is said to have the
\emph{finiteness property} if there exists a periodic product which achieves
this maximal rate of growth. J.C. Lagarias and Y. Wang conjectured in 1995 that
every finite set of real matrices satisfies the finiteness
property. However, T. Bousch and J. Mairesse proved in 2002 that
counterexamples to the finiteness conjecture exist, showing in particular that
there exists a family of pairs of matrices which contains a
counterexample. Similar results were subsequently given by V.D. Blondel, J.
Theys and A.A. Vladimirov and by V.S. Kozyakin, but no explicit counterexample
to the finiteness conjecture has so far been given. The purpose of this paper
is to resolve this issue by giving the first completely explicit description of
a counterexample to the Lagarias-Wang finiteness conjecture. Namely, for the
set \mathsf{A}_{\alpha_*}:= \{({cc}1&1\\0&1), \alpha_*({cc}1&0\\1&1)\} we
give an explicit value of \alpha_* \simeq
0.749326546330367557943961948091344672091327370236064317358024...] such that
does not satisfy the finiteness property.Comment: 27 pages, 2 figure
- …