On the fiber product of noncompact Riemann surfaces

For every pair of non-constant holomorphic maps \(\beta_i\colon S_i\to S_0\) between noncompact Riemann surfaces, where \(i\in\{1,2\}\), there exists an associated fiber product \(S_1\times_{(\beta_{1},\beta_{2})}S_2\) that has the structure of a singular Riemann surface, endowed with a canonical map \(\beta\) to \(S_0\) satisfying \(\beta_i\circ \pi_i=\beta\), where \(\pi_i\) is coordinate projection onto \(S_i\). This paper explores the relationship between the space of ends of this fiber product and the space of ends of its normal fiber product. In addition, we establish conditions on the maps \(\beta_1\) and \(\beta_2\) that ensure connectivity in the fiber product. Upon examination of these conditions, we establish a connection between the space of ends of the fiber product and the topological characteristics of the Riemann surfaces \(S_1\) and \(S_2\). Finally, we investigate the fiber product of infinite superelliptic curves by analyzing its connectedness and the space of ends.