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 ,QT , andQV , prove that QF ,QT , andQV are of type F∞, and calculate finite presentations for them. We calculate the normal subgroup structure 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.

We investigate the representation theory of finite sets. The correspondence functors are the functors from the category of finite sets and correspondences to the category of k-modules, where k is a commutative ring. They have various specific properties which do not hold for other types of func-tors. In particular, if k is a field and if F is a correspondence functor, then F is finitely generated if and only if the dimension of F (X) grows exponentially in terms of the cardinality of the finite set X. Moreover, in such a case, F has actually finite length. Also, if k is noetherian, then any subfunctor of a finitely generated functor is finitely generated.

We calculate the centralisers of elements, finite subgroups and virtually cyclic subgroups of Houghton's group Hn. We discuss various Bredon (co)homological finiteness conditions satisfied by Hn including the Bredon (co)homological dimension and FP n conditions, which are analogs of the ordinary cohomological dimension and FPn conditions respectively.

We prove that for any discrete group G with finite F-cohomological dimension, the Gorenstein cohomological dimension equals the F-cohomological dimension. This is achieved by constructing a long exact sequence of cohomological functors, analogous to that constructed by Avramov and Martsinkovsky ['Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension', Proc. London Math. Soc. (3) 85 (2002) 393-440, § 7], containing the Fcohomology and complete F-cohomology. As a corollary, we improve upon a theorem of Degrijse concerning subadditivity of the F-cohomological dimension under group extensions [Degrijse, 'Bredon cohomological dimensions for proper actions and Mackey functors', Preprint, 2013, arXiv:1305.1227].

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