On the finite presentation of subdirect products and the nature of residually free groups

We establish {\em{virtual surjection to pairs}} (VSP) as a general criterion for the finite presentability of subdirect products of groups: if $\Gamma_1,...,\Gamma_n$ are finitely presented and $S<\Gamma_1\times...\times\Gamma_n$ projects to a subgroup of finite index in each $\Gamma_i\times\Gamma_j$, then $S$ is finitely presentable, indeed there is an algorithm that will construct a finite presentation for $S$. We use the VSP criterion to characterise the finitely presented residually free groups. We prove that the class of such groups is recursively enumerable. We describe an algorithm that, given a finite presentation of a residually free group, constructs a canonical embedding into a direct product of finitely many limit groups. We solve the (multiple) conjugacy problem and membership problem for finitely presentable subgroups of residually free groups. We also prove that there is an algorithm that, given a finite generating set for such a subgroup, will construct a finite presentation. New families of subdirect products of free groups are constructed, including the first examples of finitely presented subgroups that are neither ${\rm{FP}}_\infty$ nor of Stallings-Bieri typeComment: 44 pages. To appear in American Journal of Mathematics. This is a substantial rewrite of our previous Arxiv article 0809.3704, taking into account subsequent developments, advice of colleagues and referee's comment

Finitely presented subgroups of automatic groups and their isoperimetric functions

We describe a general technique for embedding certain amalgamated products into direct products. This technique provides us with a way of constructing a host of finitely presented subgroups of automatic groups which are not even asynchronously automatic. We can also arrange that such subgroups satisfy, at best, an exponential isoperimetric inequality.Comment: DVI and Post-Script files only. To appear in J. London Math. So

Subgroups of direct products of limit groups

If $G_1,...,G_n$ are limit groups and $S\subset G_1\times...\times G_n$ is of type \FP_n(\mathbb Q) then $S$ contains a subgroup of finite index that is itself a direct product of at most $n$ limit groups. This settles a question of Sela.Comment: 20 pages, no figures. Final version. Accepted by the Annals of Mathematic

Complete embeddings of groups

Every countable group $G$ can be embedded in a finitely-generated group $G^*$ that is hopfian and {\em complete}, i.e.~$G^*$ has trivial centre and every epimorphism $G^*\to G^*$ is an inner automorphism. Every finite subgroup of $G^*$ is conjugate to a finite subgroup of $G$. If $G$ has a finite presentation (respectively, a finite classifying space), then so does $G^*$. Our construction of $G^*$ relies on the existence of closed hyperbolic 3-manifolds that are asymmetric and non-Haken

INVERSION IS POSSIBLE IN GROUPS WITH NO PERIODIC AUTOMORPHISMS

International audienceThere exist infinite, finitely presented, torsion-free groups G such that Aut(G) and Out(G) are torsion-free but G has an automorphism sending some non-trivial element to its inverse

Case Study: Attitudes of Rural High School Students and Teachers Regarding Inclusion

This case study was intended to explore the premise that the perceptions of the stakeholders regarding inclusion should enhance the implementation of the process in a k-12 rural setting. Therefore, rural high school students’ and rural general education and special education teachers’ perceptions of inclusion provided the primary focus of this case study. Data analysis identified that while overall general education teachers supported the idea of inclusion they did not believe that they were trained. Additionally, the students supported the concept of inclusion when they were allowed choice in which classroom they were placed and if the teacher allowed choice in classroom activities. Also the classroom size was identified by all stakeholders as an issue by being affected negatively by the addition of more students being placed in inclusive classrooms. Implications for the teacher training, and the allocation of resources in rural settings are significant

Ecocide, Genocide, Capitalism and Colonialism: Consequences for indigenous peoples and glocal ecosystems environments

Continuing injustices and denial of rights of indigenous peoples are part of the long legacy of colonialism. Parallel processes of exploitation and injustice can be identified in relation to non-human species and/or aspects of the natural environment. International law can address some extreme examples of the crimes and harms of colonialism through the idea and legal definition of genocide, but the intimately related notion of ecocide that applies to nature and the environment is not yet formally accepted within the body of international law. In the context of this special issue reflecting on the development of green criminology, the article argues that the concept of ecocide provides a powerful tool. To illustrate this, the article explores connections between ecocide, genocide, capitalism and colonialism and discusses impacts on indigenous peoples and on local and global (glocal) eco-systems