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Brita E. A. Nucinkis

Publications and source records attributed to Brita E. A. Nucinkis.

9 recordsLinked to original sources
arXiv2017

Quasi-automorphisms of the infinite rooted 2-edge-coloured binary tree

We study the group $QV$, the self-maps of the infinite $2$-edge coloured binary tree which preserve the edge and colour relations at cofinitely many locations. We introduce related groups $QF$, $QT$, $\tilde{Q}T$, and $\tilde{Q}V$, prove that $QF$, $\tilde{Q}T$, and $\tilde{Q}V$ are of type $\mathrm{F}_\infty$, and calculate finite presentations for them. We calculate the normal subgroup structure and rational homology of all $5$ groups, the Bieri--Neumann--Strebel--Renz invariants of $QF$, and discuss the relationship of all $5$ groups with other generalisations of Thompson's groups.

math.GR↗
arXiv2016

Presentations of generalisations of Thompson's group $V$

We consider generalisations of Thompson's group $V$, denoted by $V_r(Σ)$, which also include the groups of Higman, Stein and Brin. It was shown by the authors in [20] that under some mild conditions these groups and centralisers of their finite subgroups are of type $\mathrm{F}_\infty$. Under more general conditions we show that the groups $V_r(Σ)$ are finitely generated and, under the mild conditions mentioned above, we see that they are finitely presented and give a recipe to find explicit presentations. For the centralisers of finite subgroups we find a suitable infinite presentation and then apply a general procedure to shorten this presentation. In the appendix, we give a proof of this general shortening procedure.

math.GR↗
arXiv2014

Cohomological finiteness conditions and centralisers in generalisations of Thompson's group V

We consider generalisations of Thompson's group $V$, denoted $V_r(Σ)$, which also include the groups of Higman, Stein and Brin. We show that, under some mild hypotheses, $V_r(Σ)$ is the full automorphism group of a Cantor-algebra. Under some further minor restrictions, we prove that these groups are of type $\mathrm{F}_\infty$ and that this implies that also centralisers of finite subgroups are of type $\mathrm{F}_\infty$.

math.GR↗
arXiv2014

Complete Bredon cohomology and its applications to hierarchically defined groups

By considering the Bredon analogue of complete cohomology of a group, we show that every group in the class $\LHFF$ of type Bredon-$\FP_\infty$ admits a finite dimensional model for $\EFG$. We also show that abelian-by-infinite cyclic groups admit a $3$-dimensional model for the classifying space for the family of virtually nilpotent subgroups. This allows us to prove that for $\mF,$ the class of virtually cyclic groups, the class of $\LHFF$-groups contains all locally virtually soluble groups and all linear groups over $\mathbb C$ of integral characteristic.

math.GR↗
arXiv2013

Bredon cohomological finiteness conditions for generalisations of Thompson's groups

We define a family of groups that generalises Thompson's groups $T$ and $G$ and also those of Higman, Stein and Brin. For groups in this family we descrine centralisers of finite subgroups and show, that for a given finite subgroup $Q$, there are finitely many conjugacy classes of finite subgroups isomorphic to $Q$. We use this to show that whenever defined, the T versions of these groups have a slightly weaker property, quasi-$\underline{\operatorname F}_\infty$, to that of a group possessing a finite type model for the classifying space for proper actions ${\underline{E}}G$ if and only if they posses finite type models for the ordinary classifying space. We also generalise some well-known properties of ordinary cohomology to Bredon cohomology.

math.GR↗
arXiv2012

Some H1F-groups with unbounded torsion and a conjecture of Kropholler and Mislin

Kropholler and Mislin conjectured that groups acting admissibly on a finite-dimensional G-CW-complex with finite stabilisers admit a finite-dimensional model for E_FG, the classifying space for proper actions. This conjecture is known to hold for groups with bounded torsion. In this note we consider a large class of groups U containing the above and many known examples with unbounded torsion. We show that the conjecture holds for a large subclass of U.

math.GR↗
arXiv2009

On groups acting on contractible spaces with stabilizers of prime power order

We study actions of discrete groups on contractible topological spaces in which either (1) all stabilizers lie in the family of subgroups of prime power order or (2) all stabilizers lie in the family of finite subgroups. We compare the classifying spaces for actions with stabilizers in these two families, the Kropholler hierarchies build on these two families, and group cohomology relative to these two families.

math.GR↗