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Jong Bum Lee

Publications and source records attributed to Jong Bum Lee.

11 recordsLinked to original sources
arXiv2022

Left invariant Lorentzian metrics and curvatures on non-unimodular Lie groups of dimension three

For each connected and simply connected three-dimensional non-unimodular Lie group, we classify the left invariant Lorentzian metrics up to automorphism, and study the extent to which curvature can be altered by a change of metric. Thereby we obtain the Ricci operator, the scalar curvature, and the sectional curvatures as functions of left invariant Lorentzian metrics on the three-dimensional non-unimodular Lie groups. Our study is a continuation and extension of the previous studies done in \cite{HL2009_MN} for Riemannian metrics on three-dimensional Lie groups and in \cite{BC} for Lorentzian metrics on three-dimensional unimodular Lie groups.

math.DG↗
arXiv2016

An averaging formula for the coincidence Reidemeister trace

In the setting of continuous maps between compact orientable manifolds of the same dimension, there is a well known averaging formula for the coincidence Lefschetz number in terms of the Lefschetz numbers of lifts to some finite covering space. We state and prove an analogous averaging formula for the coincidence Reidemeister trace. This generalizes a recent formula in fixed point theory by Liu and Zhao. We give two separate and independent proofs of our main result: one using methods developed by Kim and the first author for averaging Nielsen numbers, and one using an axiomatic approach for the local Reidemeister trace. We also give some examples and state some open questions for the nonorientable case.

math.AT↗
arXiv2015

The Nielsen numbers of iterations of maps on infra-solvmanifolds of type $R$ and periodic points

We study the asymptotic behavior of the sequence of the Nielsen numbers $\{N(f^k)\}$, the essential periodic orbits of $f$ and the homotopy minimal periods of $f$ by using the Nielsen theory of maps $f$ on infra-solvmanifolds of type $R$. We give a linear lower bound for the number of essential periodic orbits of such a map, which sharpens well-known results of Shub and Sullivan for periodic points and of Babenko and Bogatyi for periodic orbits. We also verify that a constant multiple of infinitely many prime numbers occur as homotopy minimal periods of such a map.

math.DS↗
arXiv2015

Growth rate of endomorphisms of Houghton's groups

A Houghton's group $\mathcal{H}_n$ consists of translations at infinity of a $n$ rays of discrete points on the plane. In this paper we study the growth rate of endomorphisms of Houghton's groups. We show that if the kernel of an endomorphism $ϕ$ is not trivial then the growth rate $\mathrm{GR}(ϕ)$ equals either $1$ or the spectral radius of the induced map on the abelianization. It turns out that every monomorphism $ϕ$ of $\mathcal{H}_n$ determines a unique natural number $\ell$ such that $ϕ(\mathcal{H}_n)$ is generated by translations with the same translation length $\ell$. We use this to show that $\mathrm{GR}(ϕ)$ of a monomorphism $ϕ$ of $\mathcal{H}_n$ is precisely $\ell$ for all $2\leq n$.

math.GR↗
arXiv2015

The $R_{\infty}$ property for Houghton's groups

We study twisted conjugacy classes of a family of groups which are called Houghton's groups $\mathcal{H}_n$ ($n \in\mathbb{N}$), the group of translations of $n$ rays of discrete points at infinity. We prove that the Houghton's groups $\mathcal{H}_n$ have the $R_{\infty}$ property for all $n\in \mathbb{N}$.

math.GR↗
arXiv2014

Growth rate for endomorphisms of finitely generated nilpotent groups and solvable groups

We prove that the growth rate of an endomorphism of a finitely generated nilpotent group equals to the growth rate of induced endomorphism on its abelinization, generalizing the corresponding result for an automorphism in [14]. We also study growth rates of endomorphisms for specific solvable groups, lattices of Sol, providing a counterexample to a known result in [5] and proving that the growth rate is an algebraic number.

math.GR↗
arXiv2014

The $R_\infty$ property for crystallographic group of Sol

There are 9 kinds of crystallographic groups $Π$ of Sol. For any automorphism $φ$ on $Π$, we study the Reidemeister number $R(φ)$. Using the averaging formula for the Reidemeister numbers, we prove that most of the crystallographic groups of Sol have the $R_\infty$ property.

math.AT↗
arXiv2014

Density of the homotopy minimal periods of maps on infra-solvmanifolds

We study the homotopical minimal periods for maps on infra-solvmanifolds of type (R) using the density of the homotopical minimal period set in the natural numbers. This extends the result of [10] from flat manifolds to infra-solvmanifolds of type (R). Applying our main result we will list all possible maps on infra-solvmanifolds up to dimension three for which the corresponding density is positive.

math.DS↗
arXiv2014

The Nielsen and the Reidemeister numbers of maps on infra-solvmanifolds of type (R)

We prove the rationality, the functional equations and calculate the radii of convergence of the Nielsen and the Reidemeister zeta functions of continuous maps on infra-solvmanifolds of type $\R$. We find a connection between the Reidemeister and Nielsen zeta functions and the Reidemeister torsions of the corresponding mapping tori. We show that if the Reidemeister zeta function is defined for a homeomorphism on an infra-solvmanifold of type $\R$, then this manifold is an infra-nilmanifold. We also prove that a map on an infra-solvmanifold of type $\R$ induced by an affine map minimizes the topological entropy in its homotopy class and it has a rational Artin-Mazur zeta function. Finally we prove the Gauss congruences for the Reidemeister and Nielsen numbers of any map on an infra-solvmanifolds of type $\R$ whenever all the Reidemeister numbers of iterates of the map are finite. Our main technical tool is the averaging formulas for the Lefschetz, the Nielsen and the Reidemeister numbers on infra-solvmanifolds of type $\R$.

math.GR↗
arXiv2012

Topology of iterated $S^1$-bundles

In this paper we investigate what kind of manifolds arise as the total spaces of iterated $S^1$-bundles. A real Bott tower studied in \cite{CMO}, \cite{KM} and \cite{KN} is an example of an iterated $S^1$-bundle. We show that the total space of an iterated $S^1$-bundle is homeomorphic to an infra-nilmanifold. A real Bott manifold, which is the total space of a real Bott tower, provides an example of a closed flat Riemannian manifold. We also show that real Bott manifolds are the only closed flat Riemannian manifolds obtained from iterated $\bbr{P}^1$-bundles. Finally we classify the homeomorphism types of the total spaces of iterated $S^1$-bundles in dimension 3.

math.AT↗