62,323 research outputs found
Spectral rigidity of automorphic orbits in free groups
It is well-known that a point in the (unprojectivized)
Culler-Vogtmann Outer space is uniquely determined by its
\emph{translation length function} . A subset of a
free group is called \emph{spectrally rigid} if, whenever
are such that for every then in . By
contrast to the similar questions for the Teichm\"uller space, it is known that
for there does not exist a finite spectrally rigid subset of .
In this paper we prove that for if is a subgroup
that projects to an infinite normal subgroup in then the -orbit
of an arbitrary nontrivial element is spectrally rigid. We also
establish a similar statement for , provided that is not
conjugate to a power of .
We also include an appended corrigendum which gives a corrected proof of
Lemma 5.1 about the existence of a fully irreducible element in an infinite
normal subgroup of of . Our original proof of Lemma 5.1 relied on a
subgroup classification result of Handel-Mosher, originally stated by
Handel-Mosher for arbitrary subgroups . After our paper was
published, it turned out that the proof of the Handel-Mosher subgroup
classification theorem needs the assumption that be finitely generated. The
corrigendum provides an alternative proof of Lemma~5.1 which uses the
corrected, finitely generated, version of the Handel-Mosher theorem and relies
on the 0-acylindricity of the action of on the free factor complex
(due to Bestvina-Mann-Reynolds). A proof of 0-acylindricity is included in the
corrigendum.Comment: Included a corrigendum which gives a corrected proof of Lemma 5.1
about the existence of a fully irreducible element in an infinite normal
subgroup of of Out(F_N). Note that, because of the arXiv rules, the
corrigendum and the original article are amalgamated into a single pdf file,
with the corrigendum appearing first, followed by the main body of the
original articl
Palindromic primitives and palindromic bases in the free group of rank two
The present paper records more details of the relationship between primitive
elements and palindromes in F_2, the free group of rank two. We characterise
the conjugacy classes of primitive elements which contain palindromes as those
which contain cyclically reduced words of odd length. We identify large
palindromic subwords of certain primitives in conjugacy classes which contain
cyclically reduced words of even length. We show that under obvious conditions
on exponent sums, pairs of palindromic primitives form palindromic bases for
F_2. Further, we note that each cyclically reduced primitive element is either
a palindrome, or the concatenation of two palindromes.Comment: 8 page
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