Homological finiteness conditions for groups, monoids and algebras

Recently Alonso and Hermiller introduced a homological finiteness condition\break $bi{-}FP_n$ (here called {\it weak} $bi{-}FP_n$) for monoid rings, and Kobayashi and Otto introduced a different property, also called $bi{-}FP_n$ (we adhere to their terminology). From these and other papers we know that: $bi{-}FP_n \Rightarrow$ left and right $FP_n \Rightarrow$ weak $bi{-}FP_n$; the first implication is not reversible in general; the second implication is reversible for group rings. We show that the second implication is reversible in general, even for arbitrary associative algebras (Theorem 1'), and we show that the first implication {\it is} reversible for group rings (Theorem 2). We also show that the all four properties are equivalent for connected graded algebras (Theorem 4). A result on retractions (Theorem 3') is proved, and some questions are raised.Comment: 10 page

On finite complete rewriting systems, finite derivation type, and automaticity for homogeneous monoids

This paper investigates the class of finitely presented monoids defined by homogeneous (length-preserving) relations from a computational perspective. The properties of admitting a finite complete rewriting system, having finite derivation type, being automatic, and being biautomatic are investigated for this class of monoids. The first main result shows that for any consistent combination of these properties and their negations, there is a homogeneous monoid with exactly this combination of properties. We then introduce the new concept of abstract Rees-commensurability (an analogue of the notion of abstract commensurability for groups) in order to extend this result to show that the same statement holds even if one restricts attention to the class of n-ary homogeneous monoids (where every side of every relation has fixed length n). We then introduce a new encoding technique that allows us to extend the result partially to the class of n-ary multihomogenous monoids

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 $M$-equivariant homotopy theory where $M$ is a discrete monoid. For projective $M$-CW complexes we prove several fundamental results such as the homotopy extension and lifting property, which we use to prove the $M$-equivariant Whitehead theorems. We define a left equivariant classifying space as a contractible projective $M$-CW complex. We prove that such a space is unique up to $M$-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-$\mathrm{F}_n$ 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 $M$-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-$\mathrm{F}_n$ 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 $\mathrm{FP}_n$, cohomological dimension, and Hochschild cohomological dimension. We also develop the corresponding theory of $M$-equivariant collapsing schemes (that is, $M$-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-$\mathrm{FP}_\infty$.Comment: 59 pages, 1 figur

String rewriting systems and associated finiteness conditions

We begin with an introduction which describes the thesis in detail, and then a preliminary chapter in which we discuss rewriting systems, associated complexes and finiteness conditions. In particular, we recall the graph of derivations r and the 2- complex V associated to any rewriting system, and the related geometric finiteness conditions F DT and F HT. In §1.4 we give basic definitions and results about finite complete rewriting systems, that is, rewriting systems which rewrite any word in a finite number of steps to its normal form, the unique irreducible word in its congruence class. The main work of the thesis begins in Chapter 2 with some discussion of rewriting systems for groups which are confluent on the congruence class containing the empty word. In §2.1 we characterize groups admitting finite A-complete rewriting systems as those with a A-Dehn presentation, and in §2.2 we give some examples of finite rewriting systems for groups which are A-complete but not complete. For the remainder of the thesis, we study how the properties of finite complete rewriting systems which are discussed in the first chapter are mirrored in higher dimensions. In Chapter 3 we extend the 2-complex V to form a new 3-complex VP, and in Chapter 4 we define new finiteness conditions F DT2 and F HT2 based on the homotopy and homology of this complex. In §4.4 we show that if a monoid admits a finite complete rewriting system, then it is of type F DT2 • The final chapter contains a discussion of alternative ways to define such higher dimensional finiteness conditions. This leads to the introduction, in §5.2, of a variant of the Guba-Sapir homotopy reduction system which can be associated to any co~­ plete rewriting system. This is a rewriting system operating on paths in r, and is complete in the sense that it rewrites paths in a finite number of steps to a unique "normal form"