Free crossed resolutions of groups and presentations of modules of identities among relations

Quasirationality and prounipotent crossed modules

Quasirational presentations ($QR$-presentations) of (pro-$p$)groups are studied. Such presentations include, in particular, aspherical presentations of discrete groups and their subpresentations and pro-$p$-groups with a single defining relation. Using schematization of $QR$-presentations we answer a variation of J.-P. Serre's question on one-relator pro-$p$-groups.

Proalgebraic crossed modules of quasirational presentations

We introduce the concept of quasirational relation modules for discrete and pro-$p$ presentations of discrete and pro-$p$ groups and show that aspherical presentations and their subpresentations are quasirational. In the pro-$p$-case quasirationality of pro-$p$-groups with a single defining relation holds. For every quasirational (pro-$p$)relation module we construct the so called $p$-adic rationalization, which is a pro-fd-module $\overline{R}\widehat{\otimes}\mathbb{Q}_p= \varprojlim R/[R,R\mathcal{M}_n]\otimes\mathbb{Q}_p$. We provide the isomorphisms $\overline{R^{\wedge}_w}(\mathbb{Q}_p)=…

交叉模

Crossed modules

本文综述了交叉模的概念，指出其由群、自同构群及典范同态构成，并源于群展示关系、扩张问题、低维拓扑及非交换代数推广，且对Lie代数同样适用。

Semigroup presentations for test local groups

In this paper we exhibit a type of semigroup presentations which determines a class of local groups. We show that the finite elements of this class generate the pseudovariety ${\bf LG}$ of all finite local groups and use them as test-semigroups to prove that ${\bf LG}$ and ${\bf S}$, the pseudovariety of all finite semigroups, verify the same $κ$-identities involving $κ$-terms of rank at most 1, where $κ$ denotes the implicit signature consisting of the multiplication and the $(ω-1)$-power.

The algebra of rewriting for presentations of inverse monoids

We describe a formalism, using groupoids, for the study of rewriting for presentations of inverse monoids, that is based on the Squier complex construction for monoid presentations. We introduce the class of pseudoregular groupoids, an example of which now arises as the fundamental groupoid of our version of the Squier complex. A further key ingredient is the factorisation of the presentation map from a free inverse monoid as the composition of an idempotent pure map and an idempotent separating map. The relation module of a presentation is then defined as the abelianised kernel of this idempo…

Hecke algebras for protonormal groups

We introduce the term "protonormal" to refer to a subgroup H of a group G such that for every x in G the subgroups x^{-1}Hx and H commute as sets. If moreover (G,H) is a Hecke pair we show that the Hecke algebra H(G,H) is generated by the range of a canonical partial representation of G vanishing on H. As a consequence we show that there exists a maximum C*-norm on H(G,H), generalizing previous results by Brenken, Hall, Laca, Larsen, Kaliszewski, Landstad and Quigg. When there exists a normal subgroup N of G, containing H as a normal subgroup, we prove a new formula for the product of the gene…