Equations in Products of Free Groups and 3-Manifold Groups II: Olshanskii Epimorphisms

Abstract

In 1989 Olshanskii introduced a three-parameter family of coordinate-surjective homomorphisms from the genus-two surface group to a direct product of two rank-two free groups. When the common quotient of the two coordinate images is finite of order $n$, restriction to the corresponding regular cover produces an epimorphism \[ \pi_1(S_{n+1})\longrightarrow F_{n+1}\times F_{n+1}. \] We call these maps \emph{Olshanskii epimorphisms}. They form an explicit high-genus test family for the standardness problem for splitting epimorphisms. We prove that every Olshanskii epimorphism is standard. The genus-two homomorphism determines a Heegaard splitting of a Seifert fibered $3$-manifold over $S^2$ with at most three exceptional fibers. In the finite-quotient cases, classical Seifert theory shows that its universal cover is $S^3$; Waldhausen's theorem then implies that the lifted genus-$(n+1)$ Heegaard splitting is standard. The proof uses neither Perelman's theorem nor the general Poincar\'e theorem. We also give a constructive treatment of the quaternion case $Q(2,2,2)\cong Q_8$, whose covering surface has genus nine. A maximal tree in the quaternion Schreier graph yields the covering handlebody and explicit Schreier bases. Using geometric longitude--meridian pairs and explicit surface automorphisms supported on the nine one-holed tori, we transform the lifted meridian words into a free basis. A separate fixed-rank Andrews--Curtis certificate reduces the associated balanced presentation. This paper supplies the detailed proof of the result announced in the previous Kharlampovich, Vdovina paper.