• 대한수학회보
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• 제54권6호
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• pp.1927-1933
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• 2017

We prove that there are rational homology balls $B_p$ smoothly embedded in the 2-handlebodies associated to certain knots. Furthermore we show that, if we rationally blow up the 2-handlebody along the embedded rational homology ball $B_p$, then the resulting 4-manifold cannot be obtained just by a sequence of ordinary blow ups from the 2-handlebody under a certain mild condition.

• Kyungpook Mathematical Journal
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• 제56권2호
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• pp.577-582
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• 2016

In this paper we consider all orientation-preserving ${\mathbb{Z}}_4$-actions on 3-dimensional handlebodies $V_g$ of genus g > 0. We study the graph of groups (${\Gamma}(v)$, G(v)), which determines a handlebody orbifold $V({\Gamma}(v),G(v)){\simeq}V_g/{\mathbb{Z}}_4$. This algebraic characterization is used to enumerate the total number of ${\mathbb{Z}}_4$ group actions on such handlebodies, up to equivalence.

• Kyungpook Mathematical Journal
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• 제58권3호
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• pp.573-581
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• 2018

In this paper we consider all orientation-preserving ${\mathbb{Z}}_{p^2}$-actions on 3-dimensional handlebodies $V_g$ of genus g > 0 for p an odd prime. To do so, we examine particular graphs of groups (${\Gamma}(v)$, G(v)) in canonical form for some 5-tuple v = (r, s, t, m, n) with r + s + t + m > 0. These graphs of groups correspond to the handlebody orbifolds V (${\Gamma}(v)$, G(v)) that are homeomorphic to the quotient spaces $V_g/{\mathbb{Z}}_{p^2}$ of genus less than or equal to g. This algebraic characterization is used to enumerate the total number of ${\mathbb{Z}}_{p^2}$-actions on such handlebodies, up to equivalence.

• 대한수학회보
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• 제49권2호
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• pp.411-415
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• 2012

In 1995 it was proved by Gonz$\acute{a}$lez-Diez that the cyclic group generated by a p-gonal automorphism of a closed Riemann surface of genus at least two is unique up to conjugation in the full group of conformal automorphisms. Later, in 2008, Gromadzki provided a different and shorter proof of the same fact using the Castelnuovo-Severi theorem. In this paper we provide another proof which is shorter and is just a simple use of Sylow's theorem together with the Castelnuovo-Severi theorem. This method permits to obtain that the cyclic group generated by a conformal automorphism of order p of a handlebody with a Kleinian structure and quotient the three-ball is unique up to conjugation in the full group of conformal automorphisms.