396 research outputs found
Monoids of O-type, subword reversing, and ordered groups
We describe a simple scheme for constructing finitely generated monoids in
which left-divisibility is a linear ordering and for practically investigating
these monoids. The approach is based on subword reversing, a general method of
combinatorial group theory, and connected with Garside theory, here in a
non-Noetherian context. As an application we describe several families of
ordered groups whose space of left-invariant orderings has an isolated point,
including torus knot groups and some of their amalgamated products.Comment: updated version with new result
Almost Every Simply Typed Lambda-Term Has a Long Beta-Reduction Sequence
It is well known that the length of a beta-reduction sequence of a simply
typed lambda-term of order k can be huge; it is as large as k-fold exponential
in the size of the lambda-term in the worst case. We consider the following
relevant question about quantitative properties, instead of the worst case: how
many simply typed lambda-terms have very long reduction sequences? We provide a
partial answer to this question, by showing that asymptotically almost every
simply typed lambda-term of order k has a reduction sequence as long as
(k-1)-fold exponential in the term size, under the assumption that the arity of
functions and the number of variables that may occur in every subterm are
bounded above by a constant. To prove it, we have extended the infinite monkey
theorem for strings to a parametrized one for regular tree languages, which may
be of independent interest. The work has been motivated by quantitative
analysis of the complexity of higher-order model checking
A note on palindromicity
Two results on palindromicity of bi-infinite words in a finite alphabet are
presented. The first is a simple, but efficient criterion to exclude
palindromicity of minimal sequences and applies, in particular, to the
Rudin-Shapiro sequence. The second provides a constructive method to build
palindromic minimal sequences based upon regular, generic model sets with
centro-symmetric window. These give rise to diagonal tight-binding models in
one dimension with purely singular continuous spectrum.Comment: 12 page
Convex cocompactness in mapping class groups via quasiconvexity in right-angled Artin groups
We characterize convex cocompact subgroups of mapping class groups that arise
as subgroups of specially embedded right-angled Artin groups. That is, if the
right-angled Artin group G in Mod(S) satisfies certain conditions that imply G
is quasi-isometrically embedded in Mod(S), then a purely pseudo-Anosov subgroup
H of G is convex cocompact in Mod(S) if and only if it is combinatorially
quasiconvex in G. We use this criterion to construct convex cocompact subgroups
of Mod(S) whose orbit maps into the curve complex have small Lipschitz
constants.Comment: 30 pages, 4 figure
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