9,798 research outputs found
Abelian-Square-Rich Words
An abelian square is the concatenation of two words that are anagrams of one
another. A word of length can contain at most distinct
factors, and there exist words of length containing distinct
abelian-square factors, that is, distinct factors that are abelian squares.
This motivates us to study infinite words such that the number of distinct
abelian-square factors of length grows quadratically with . More
precisely, we say that an infinite word is {\it abelian-square-rich} if,
for every , every factor of of length contains, on average, a number
of distinct abelian-square factors that is quadratic in ; and {\it uniformly
abelian-square-rich} if every factor of contains a number of distinct
abelian-square factors that is proportional to the square of its length. Of
course, if a word is uniformly abelian-square-rich, then it is
abelian-square-rich, but we show that the converse is not true in general. We
prove that the Thue-Morse word is uniformly abelian-square-rich and that the
function counting the number of distinct abelian-square factors of length
of the Thue-Morse word is -regular. As for Sturmian words, we prove that a
Sturmian word of angle is uniformly abelian-square-rich
if and only if the irrational has bounded partial quotients, that is,
if and only if has bounded exponent.Comment: To appear in Theoretical Computer Science. Corrected a flaw in the
proof of Proposition
Words with the Maximum Number of Abelian Squares
An abelian square is the concatenation of two words that are anagrams of one
another. A word of length can contain distinct factors that
are abelian squares. We study infinite words such that the number of abelian
square factors of length grows quadratically with .Comment: To appear in the proceedings of WORDS 201
Abelian subgroup structure of square complex groups and arithmetic of quaternions
A square complex is a 2-complex formed by gluing squares together. This
article is concerned with the fundamental group of certain square
complexes of nonpositive curvature, related to quaternion algebras. The abelian
subgroup structure of is studied in some detail.Comment: 13 page
Nori 1-motives
Let EHM be Nori's category of effective homological mixed motives. In this
paper, we consider the thick abelian subcategory EHM_1 generated by the i-th
relative homology of pairs of varieties for i = 0,1. We show that EHM_1 is
naturally equivalent to the abelian category M_1 of Deligne 1-motives with
torsion; this is our main theorem. Along the way, we obtain several interesting
results. Firstly, we realize M_1 as the universal abelian category obtained,
using Nori's formalism, from the Betti representation of an explicit diagram of
curves. Secondly, we obtain a conceptual proof of a theorem of Vologodsky on
realizations of 1-motives. Thirdly, we verify a conjecture of Deligne on
extensions of 1-motives in the category of mixed realizations for those
extensions that are effective in Nori's sense
Groups whose word problems are not semilinear
Suppose that G is a finitely generated group and W is the formal language of
words defining the identity in G. We prove that if G is a nilpotent group, the
fundamental group of a finite volume hyperbolic three-manifold, or a
right-angled Artin group whose graph lies in a certain infinite class, then W
is not a multiple context free language
On the cubical geometry of Higman's group
We investigate the cocompact action of Higman's group on a CAT(0) square
complex associated to its standard presentation. We show that this action is in
a sense intrinsic, which allows for the use of geometric techniques to study
the endomorphisms of the group, and show striking similarities with mapping
class groups of hyperbolic surfaces, outer automorphism groups of free groups
and linear groups over the integers. We compute explicitly the automorphism
group and outer automorphism group of Higman's group, and show that the group
is both hopfian and co-hopfian. We actually prove a stronger rigidity result
about the endomorphisms of Higman's group: Every non-trivial morphism from the
group to itself is an automorphism. We also study the geometry of the action
and prove a surprising result: Although the CAT(0) square complex acted upon
contains uncountably many flats, the Higman group does not contain subgroups
isomorphic to Z^2. Finally, we show that this action possesses features
reminiscent of negative curvature, which we use to prove a refined version of
the Tits alternative for Higman's group.Comment: Accepted versio
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