Low dimensional homotopy for monoids II: groups

A finite rewriting system is presented that does not satisfy the homotopical finiteness condition FDT, although it satisfies the homological finiteness condition FHT. This system is obtained from a group G and a finitely generated subgroup H of G through a monoid extension that is almost an HNN-extension. The FHT property of the extension is closely related to the FP 2 property for the subgroup H, while the FDT property of the extension is related to the finite presentability of H. The example system separating the FDT property from the FHT property is then obtained by applying this construction to an example group considered by Bestvina and Brady (1997).

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“…Clearly FDT implies FHT. It is shown in [CO,Pr99] that for groups the two properties are equivalent.…”

Section: The Squier Complexmentioning

confidence: 99%

For Rewriting Systems the Topological Finiteness Conditions FDT and FHT Are Not Equivalent

Pride

Otto

2004

J. London Math. Soc.

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“…D Note that in fact any vertex group TTI(K + , [to]) of n is isomorphic to ^(L), since the right action of [w] € F on n\(K + , 1) carries it to wi(K + , [w]). This is observed by Pride in [11].…”

Section: Monoid Presentations Of Groupsmentioning

confidence: 60%

“…The first homology of if is studied by Pride [10,11] as an analogue of the module of identities among the relations in a group presentation.…”

Section: Introductionmentioning

confidence: 99%

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Monoid Presentations and Associated Groupoids

Gilbert

1998

Int. J. Algebra Comput.

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We consider properties of a 2-complex associated by Squier to a monoid presentation. We show that the fundamental groupoid admits a monoid structure, and we establish a relationship between its group completion and the fundamental group of the 2-complex. We also treat a modified complex, due to Pride, for monoid presentations of groups, and compute the structure of the fundamental groupoid in this setting. IntroductionThere now exists a well-developed theory of the low-dimensional topology of monoid presentations, the cornerstone of which is a 2-complex K := K(V) associated to a monoid presentation V := [X : R), and introduced by Squier [12]. The vertices of K are words in the free monoid X* on X, and 1-cells correspond to the application of a relation from R. A path in K thus represents a derivation of the equality of two words in the monoid M presented by V. The 2-cells of K are attached so as to make a path consisting of the application of two relations to non-overlapping segments of a word homotopic to their application in the reverse order.The fundamental groups of K are the diagram groups of V studied by Guba and Sapir [4], who have extensive results on the properties of diagram groups. The first homology of if is studied by Pride [10,11] as an analogue of the module of identities among the relations in a group presentation.In this paper we consider the relationship of the low-dimensional topology of monoid presentations to the theory for group presentations. In Sec. 2 we consider the operation of group completion of monoids, which takes us from monoid presentations to group presentations. We show that the fundamental groupoid n = TT(K,X*) of K admits a monoid structure defined internally in the category of groupoids, and we compute its group completion. In Sec. 3 we follow Pride [11] in considering monoid presentations of groups. Suppose we have a group presentation (X : R)

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“…The two properties F P 3 and F DT are equivalent in the class of groups [6] (see also [22]). Also, the property F P 3 is preserved when passing from a group to some finite index subgroup, and vice-versa.…”

Section: Introductionmentioning

confidence: 99%

On Finite Complete Presentations and Exact Decompositions of Semigroups

Araújo

Malheiro

2011

Communications in Algebra

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Abstract. We prove that given a finite (zero) exact right decomposition (M, T ) of a semigroup S, if M is defined by a finite complete presentation then S is also defined by a finite complete presentation. Exact right decompositions are natural generalizations to semigroups of coset decompositions in groups. As a consequence we deduce that the Zappa-Szép extension of a monoid defined by a finite complete presentation, by a finite monoid is also defined by such a presentation.It is also shown that when a semigroup A isomorphic to a variant semigroup A(x) that is defined by a finite complete presentation, where x belongs to a sandwich matrix P , together with some other conditions, we deduce that the zero Rees matrix semigroup M 0 [A; I, J; P ] is also defined by a finite complete presentation.

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Low dimensional homotopy for monoids II: groups

Cited by 19 publications

References 10 publications

For Rewriting Systems the Topological Finiteness Conditions FDT and FHT Are Not Equivalent

For Rewriting Systems the Topological Finiteness Conditions FDT and FHT Are Not Equivalent

Monoid Presentations and Associated Groupoids

On Finite Complete Presentations and Exact Decompositions of Semigroups

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