• Journal of the Korean Mathematical Society
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• v.43 no.5
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• pp.1115-1128
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• 2006

Let $so(3)\tilde{\times}R^3$ be the 6-dimensional nilpotent Lie group with group operation (s, x)(t, y) = $(s+t+xy^t-yx^t,\;x+y)$. We prove that the maximal order of the holonomy groups of all infra-nilmanifolds with $so(3)\tilde{\times}R^3$-geometry is 16.

• Journal of the Korean Mathematical Society
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• v.36 no.1
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• pp.1-13
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• 1999

Let G be the 3-dimensional Heisenberg group. A discrete subgroup of Isom(G), acting freely on G with non-compact quotient, must be isomorphic to either 1, Z, Z2 or the fundamental group of the Klein bottle. We classify all discrete representations of such groups into Isom(G) up to affine conjugacy. This yields an affine calssification of 3-dimensional non-compact infra-nilmanifolds.

• Journal of the Korean Mathematical Society
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• v.38 no.5
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• pp.883-914
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• 2001

We present a survey of research results obtained for infra-nilmanifolds, their fundamental groups and some of their generalizations. This is presented from two different approaches and covers achievements obtained during the past four decades and showing a remarkable amount of mathematical interdisciplinarity. We go more in depth concerning the existence and construction of polynomial structures for these manifolds and groups, a direction where significant progress was made in the past few years. The bounded-degree polynomial structures developed by the authors triggered a number of challenging open problems. Also, their study already has lead to some interesting results concerning e.g. Anosov diffeomorphisms and expanding maps.

• Journal of the Korean Mathematical Society
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• v.38 no.6
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• pp.1107-1116
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• 2001

In this paper, we study the rationality of the Reidemeister zeta function of an endomorphism of a group extension. As an application, we give sufficient conditions for the rationality of the Reidemeister and the Nielsen zeta functions of selfmaps on an exponential solvmanifold or an infra-nilmanifold or the coset space of a compact connected Lie group by a finite subgroup.

• Journal of the Korean Mathematical Society
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• v.45 no.5
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• pp.1483-1503
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• 2008

Let f : M ${\rightarrow}$ M be a self-map on the Klein bottle M. We compute the Lefschetz number and the Nielsen number of f by using the infra-nilmanifold structure of the Klein bottle and the averaging formulas for the Lefschetz numbers and the Nielsen numbers of maps on infra-nilmanifolds. For each positive integer n, we provide an explicit algorithm for a complete computation of the Nielsen type numbers $NP_n(f)$ and $N{\Phi}_{n}(f)\;of\;f^{n}$.