4,395 research outputs found
Homology and closure properties of autostackable groups
Autostackability for finitely presented groups is a topological property of
the Cayley graph combined with formal language theoretic restrictions, that
implies solvability of the word problem. The class of autostackable groups is
known to include all asynchronously automatic groups with respect to a
prefix-closed normal form set, and all groups admitting finite complete
rewriting systems. Although groups in the latter two classes all satisfy the
homological finiteness condition , we show that the class of
autostackable groups includes a group that is not of type . We also show
that the class of autostackable groups is closed under graph products and
extensions.Comment: 20 page
Topological finiteness properties of monoids. Part 1: Foundations
We initiate the study of higher dimensional topological finiteness properties
of monoids. This is done by developing the theory of monoids acting on CW
complexes. For this we establish the foundations of -equivariant homotopy
theory where is a discrete monoid. For projective -CW complexes we prove
several fundamental results such as the homotopy extension and lifting
property, which we use to prove the -equivariant Whitehead theorems. We
define a left equivariant classifying space as a contractible projective -CW
complex. We prove that such a space is unique up to -homotopy equivalence
and give a canonical model for such a space via the nerve of the right Cayley
graph category of the monoid. The topological finiteness conditions
left- and left geometric dimension are then defined for monoids
in terms of existence of a left equivariant classifying space satisfying
appropriate finiteness properties. We also introduce the bilateral notion of
-equivariant classifying space, proving uniqueness and giving a canonical
model via the nerve of the two-sided Cayley graph category, and we define the
associated finiteness properties bi- and geometric dimension. We
explore the connections between all of the these topological finiteness
properties and several well-studied homological finiteness properties of
monoids which are important in the theory of string rewriting systems,
including , cohomological dimension, and Hochschild
cohomological dimension. We also develop the corresponding theory of
-equivariant collapsing schemes (that is, -equivariant discrete Morse
theory), and among other things apply it to give topological proofs of results
of Anick, Squier and Kobayashi that monoids which admit presentations by
complete rewriting systems are left-, right- and bi-.Comment: 59 pages, 1 figur
Higher-dimensional normalisation strategies for acyclicity
We introduce acyclic polygraphs, a notion of complete categorical cellular
model for (small) categories, containing generators, relations and
higher-dimensional globular syzygies. We give a rewriting method to construct
explicit acyclic polygraphs from convergent presentations. For that, we
introduce higher-dimensional normalisation strategies, defined as homotopically
coherent ways to relate each cell of a polygraph to its normal form, then we
prove that acyclicity is equivalent to the existence of a normalisation
strategy. Using acyclic polygraphs, we define a higher-dimensional homotopical
finiteness condition for higher categories which extends Squier's finite
derivation type for monoids. We relate this homotopical property to a new
homological finiteness condition that we introduce here.Comment: Final versio
Finite convergent presentations of plactic monoids for semisimple lie algebras
We study rewriting properties of the column presentation of plactic monoid
for any semisimple Lie algebra such as termination and confluence. Littelmann
described this presentation using L-S paths generators. Thanks to the shapes of
tableaux, we show that this presentation is finite and convergent. We obtain as
a corollary that plactic monoids for any semisimple Lie algebra satisfy
homological finiteness properties
Hopfian and co-hopfian subsemigroups and extensions
This paper investigates the preservation of hopficity and co-hopficity on
passing to finite-index subsemigroups and extensions. It was already known that
hopficity is not preserved on passing to finite Rees index subsemigroups, even
in the finitely generated case. We give a stronger example to show that it is
not preserved even in the finitely presented case. It was also known that
hopficity is not preserved in general on passing to finite Rees index
extensions, but that it is preserved in the finitely generated case. We show
that, in contrast, hopficity is not preserved on passing to finite Green index
extensions, even within the class of finitely presented semigroups. Turning to
co-hopficity, we prove that within the class of finitely generated semigroups,
co-hopficity is preserved on passing to finite Rees index extensions, but is
not preserved on passing to finite Rees index subsemigroups, even in the
finitely presented case. Finally, by linking co-hopficity for graphs to
co-hopficity for semigroups, we show that without the hypothesis of finite
generation, co-hopficity is not preserved on passing to finite Rees index
extensions.Comment: 15 pages; 3 figures. Revision to fix minor errors and add summary
table
- …