Thompson’s groups \(F\), \(T\) and \(G\) may be defined as automorphism groups of certain Cantor algebras. Using specific Cantor algebras, denoted by \(U_r(\Sigma)\), there are generalizations \(G_r(\Sigma)\) and \(T_r(\Sigma)\) of \(G\) and \(T\). The groups \(G_r(\Sigma)\) are defined as bijections of admissible subsets of \(U_r(\Sigma)\) and the groups \(T_r(\Sigma)\) are bijections that preserve the cyclic order. The admissible subsets of \(U_r(\Sigma)\) form a poset and the groups \(T_r(\Sigma)\) and \(G_r(\Sigma)\) act on its geometric realization \(|\mathcal U_r(\Sigma)|\).
The authors prove the following property of this space: Theorem. The space \(|\mathcal U_r(\Sigma)|\) is a model for the classifying space for proper actions for \(G_r(\Sigma)\).
The authors explore finiteness conditions in Bredon cohomology for these groups. More precisely, a group, \(G\), is of type quasi-\(\underline{\text{FP}}_\infty\) if \(G\) is of type Bredon-\(\text{FP}_\infty\), the centralizers of its finite subgroups are finitely presented and for each subgroup \(Q\subset G\) there are only finitely many conjugacy classes of subgroups isomorphic to \(Q\). The authors prove the following Theorem. The group \(T_r(\Sigma)\) is of type quasi-\(\underline{\text{FP}}_\infty\) if and only if \(T_s(\Sigma)\) is of type \(\text{FP}_\infty\) for each \(1\leq s\leq d\) with \(\gcd(s,d)\mid r\).