Epimorphism sequences between hyperbolic 3-manifold groups (Q2781274)

The author shows that any infinite sequence of epimorphisms between finitely generated hyperbolic 3-manifold groups eventually consists of isomorphisms. This result stands in contrast to the example cited by the author from Reid-Wang-Zhou of a closed hyperbolic 3-manifold \(M\) which admits, for any \(n\in \mathbb{N}\), a length \(n\) sequence \(M_0 @>f_0>> M_1 @>f_1>> \cdots @>f_{n-1}>> M_n\) of non-homotopy equivalence, \(\pi_1\)-surjective maps between closed hyperbolic 3-manifolds \(M_i\) \((i=0,1, \dots,n)\) with \(M_0=M\). The author's proof analyzes the character variety \(X(G)\) of representations of a finitely generated group \(G\) into \(SL_2(Z)\) by counting, for each \(i,\) the number of \(i\)-dimensional irreducible components of \(X(G)\).