Basic partitions and combinations of group actions on the circle: A new approach to a theorem of Kathryn Mann (Q515300)

Let \(S^1= \mathbb{R}/\mathbb{Z}\) and let \(\pi:\mathbb{R} \rightarrow S^1\) be the canonical projection. The group of orientation-preserving homeomorphisms of \(S^1\) is denoted by \(\mathcal{H}\), and \(\mathcal{R}_G\) denotes the set of the homomorphisms from \(G\) to \(\mathcal{H}\). The author defines the set \[ \mathcal{R}_G^* =\{ \varphi \in \mathcal{R}_G : \exists \;\; x \in S^1\text{ such that }\varphi(g)(x)=x,\;\forall g \in G\}. \] Given two homomorphisms \(\varphi_1\) and \(\varphi_2\) in \(\mathcal{R}_G\), they are said to be semiconjugate, \(\varphi_1\equiv\varphi_2\), if either \(\varphi_1\) and \(\varphi_2\) are in \(\mathcal{R}_G^*\) or \(\varphi_1\) and \(\varphi_2\) are in \(\mathcal{R}_G \setminus \mathcal{R}_G^*\) and there exists a degree-one monotone map \(h : S^1 \rightarrow S^1\) (i.e., a nondecreasing map \(\tilde{h}:\mathbb{R} \rightarrow \mathbb{R}\) such that \(\tilde{h} \circ T=T\circ \tilde{h}\) and \(\pi \circ \tilde{h} = h \circ \pi\)) such that \(\varphi_2 (g) \circ h = h \circ \varphi_1(g)\) for every \(g \in G\). The relation \(\equiv \) is an equivalence relation. A homomorphism \(\varphi \in \mathcal{R}_G \) is of type-0 if there exists a \(\varphi (G)\)-invariant probability measure on \(S^1\). More generally, a homomorphism \(\varphi \in \mathcal{R}_G\) is of type-\(k\), \(k \in \mathbb N\), if it satisfies the conditions: (1) \(\varphi\) is not of type-0. (2) The minimal set of \(\varphi\) is a Cantor set and a homomorphism \(\varphi_{\#}\) semiconjugate to \(\varphi\) is a \(k\)-fold lift of some homomorphism in \(\mathcal{R}_G.\) (3) \(k\) is the maximal among those which satisfy (2). The set of type-\(k\) homomorphisms is denoted by \(\mathcal{R}_G (k)\). The group \(\mathcal{H}\) is a topological group with the uniform convergence topology \[ d(f,h) = \sup_{x \in S^1} |f(x) -h(x)|,\text{ for }f, h \in \mathcal{H}. \] The space \(\mathcal{R}_G\) is endowed with the weak topology. Given \(\varphi \in \mathcal{R}_G\), \(g \in G\) and \(\epsilon >0\), let \[ U(\varphi; g, \epsilon)=\{ \psi \in \mathcal{R}_G: \, d(\psi (g), \varphi (g)) <\epsilon \}. \] The considered topology is generated by the subbase consisting of all sets \(U(\varphi; g, \epsilon)\). The author proves that for any group \(G\) and \(k \geq 1\), \(\mathcal{R}_G(0)\) is closed and \(\cup_{1\leq i \leq k} \mathcal{R}_G(i)\) is open in \(\mathcal{R}_G\). The fundamental group \(\Pi_g\) of a surface of genus \(g\geq 2\) has the representation \[ \Pi_g =\langle A_1, B_1, \ldots , A_g, B_g| [A_1,B_1] \ldots[A_g,B_g]=e\rangle . \] Given \(\varphi \in \mathcal{R}_{\Pi_g}\), its Euler number \(\mathrm{eu}(\varphi ) \in \mathbb{Z}\) is defined by \[ [\widetilde{\varphi(A_1)},\widetilde{\varphi(B_1)}] \ldots [\widetilde{\varphi(A_g)},\widetilde{\varphi(B_g)}]]=T^{\mathrm{eu}(\varphi)}, \] with \(T:\mathbb{R} \rightarrow \mathbb{R}\) denoting the translation by 1, an arbitrary lift to \(\mathbb{R}\). This paper contains several results on the function \(\mathrm{eu}: \mathcal{R}_{\Pi_g} \rightarrow \mathbb{Z}\) and leads to a new proof of the theorem of \textit{K. Mann} [``Components of surface group representations'', Preprint, \url{arXiv:1309.2905}].