In this paper we study analogues of amenability for topological groups in the context of definable structures. We prove fixed point theorems for such groups. More importantly, we propose definitions …
In this paper we study analogues of amenability for topological groups in the context of definable structures. We prove fixed point theorems for such groups. More importantly, we propose definitions for definable actions and continuous functions from definable groups to topological spaces which might prove useful in other contexts.
Let $G$ be a finitely generated group acting on a compact Hausdorff space ${\mathcal{X}}$ . We give a fixed point characterisation for the action being amenable. As a corollary, we …
Let $G$ be a finitely generated group acting on a compact Hausdorff space ${\mathcal{X}}$ . We give a fixed point characterisation for the action being amenable. As a corollary, we obtain a fixed point characterisation for the exactness of $G$ .
We generalize the notions of definable amenability and extreme definable amenability to continuous structures and show that the stable and ultracompact groups are definable amenable. In addition, we characterize both …
We generalize the notions of definable amenability and extreme definable amenability to continuous structures and show that the stable and ultracompact groups are definable amenable. In addition, we characterize both notions in terms of fixed-point properties. We prove that, for dependent theories, definable amenability is equivalent to the existence of a good S1 ideal. Finally, we show the randomizations of first-order definable amenable groups are extremely definably amenable.