Author:
Gray Robert D.,Steinberg Benjamin
Abstract
AbstractFor every one-relator monoid $$M = \langle A \mid u=v \rangle $$
M
=
⟨
A
∣
u
=
v
⟩
with $$u, v \in A^*$$
u
,
v
∈
A
∗
we construct a contractible M-CW complex and use it to build a projective resolution of the trivial module which is finitely generated in all dimensions. This proves that all one-relator monoids are of type $$\mathrm{FP}_{\infty }$$
FP
∞
, answering positively a problem posed by Kobayashi in 2000. We also apply our results to classify the one-relator monoids of cohomological dimension at most 2, and to describe the relation module, in the sense of Ivanov, of a torsion-free one-relator monoid presentation as an explicitly given principal left ideal of the monoid ring. In addition, we prove the topological analogues of these results by showing that all one-relator monoids satisfy the topological finiteness property $$\mathrm{F}_\infty $$
F
∞
, and classifying the one-relator monoids with geometric dimension at most 2. These results give a natural monoid analogue of Lyndon’s Identity Theorem for one-relator groups.
Publisher
Springer Science and Business Media LLC
Subject
General Physics and Astronomy,General Mathematics
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