Topological dynamics for groups definable in real closed field (Q487688)

This paper gives an extension of results by \textit{G. Jagiella} [Math. Log. Q. 61, No. 1--2, 45--55 (2015; Zbl 1341.03049)]. The main theorem of this paper says that for a group \(G\) definable over an o-minimal expansion of the reals and having a decomposition \(G=HK\), \(K\) definably compact and \(H\) torsion free, and for an elementary extension \(M\) of \(\mathbb{R}\), the Ellis group of the Ellis semigroup \(S_G(M^{\mathrm{ext}})\) of \(S_G(M)\), where \(S_G(M)\) is the space of types of \(G\) over \(M\), is algebraically isomorphic to \((N_G(H) \cap K) (\mathbb{R})\). Here, \(S_G(M^{\mathrm{ext}})\) is the space of external types of \(G\) over \(M\). This leads the authors to extend a counter-example to a conjecture of Newelski saying that for a theory with NIP, the Ellis groups of the Ellis semigroup (the universal definable \(G(M)\)-flow) \(S_G(M^{\mathrm{ext}})\) of the \(G(M)\)-flow \(S_G(M)\) are isomorphic to \(G/G^{00}\). The counter-example is \(\mathrm{SL}(2,M)\) with \(M\) an elementary extension of the reals, not just the reals. The case \(M\neq \mathbb{R}\) is shown to be different from the case \(M=\mathbb{R}\).