Kyung Bai Lee

Averaging formula for Nielsen numbers of maps on infra-solvmanifolds of type (R)

We prove that the averaging formula for Nielsen numbers holds for continuous maps on infra-solvmanifolds of type (R): Let M be such a manifold with holonomy group Ψ and let f: M → M be a continuous map. The averaging formula for Nielsen numbers is proved. This is a workable formula for the difficult number N(f) .

AVERAGING FORMULA FOR NIELSEN NUMBERS

We prove that the averaging formula for Nielsen numbers holds for continuous maps on infra-solvmanifolds of type (R): Let M be such a manifold with holonomy group and let f : M ! M be a continuous map. The averaging formula for Nielsen numbers N(f) = 1 j j X A2 j det(A f )j

Polynomial structures for nilpotent groups

If a polycyclic-by-finite rank-K group Γ admits a faithful affine representation making it acting on RK properly discontinuously and with compact quotient, we say that Γ admits an affine structure. In 1977, John Milnor questioned the existence of affine structures for such groups Γ. Very recently examples have been obtained showing that, even for torsion-free, finitely generated nilpotent groups N , affine structures do not always exist. It looks natural to consider affine structures as examples of polynomial structures of degree one. We introduce the concept of a canonical type polynomial structure for polycyclic-by-finite groups. Using the algebraic framework of the Seifert Fiber Space construction and a nice cohomology vanishing theorem, we prove the existence and uniqueness (up to conjugation) of canonical type polynomial structures for virtually finitely generated nilpotent groups. Applying this uniqueness to a result going back to Mal′cev, it follows that, for torsion-free, finitely generated nilpotent groups, each canonical polynomial structure is expressed in polynomials of limited degree. The minimal degree needed for obtaining a polynomial structure will determine the “affine defect number”. We prove that the known counterexamples to Milnor’s question have the smallest possible affine defect, i.e. affine defect number equal to one.

Expanding maps on infra-nilmanifolds of homogeneous type

In this paper we investigate expanding maps on infra-nilmanifolds. Such manifolds are obtained as a quotient E\L, where L is a connected and simply connected nilpotent Lie group and E is a torsion-free uniform discrete subgroup of L×C, with C a compact subgroup of Aut(L). We show that if the Lie algebra of L is homogeneous (i.e., graded and generated by elements of degree 1), then the corresponding infra-nilmanifolds admit an expanding map. This is a generalization of the result of H. Lee and K. B. Lee, who treated the 2-step nilpotent case.

Free actions of finite Abelian groups on the 3-torus

Abstract We classify free actions of finite Abelian groups on the 3-torus, up to topological conjugacy. By the works of Bieberbach and Waldhausen, this classification problem is reduced to classifying all normal Abelian subgroups of Bieberbach groups of finite index, up to affine conjugacy. All such actions are completely classified, see Theorems 2.1, 3.4, 4.2, 5.1, 6.1 and 7.1.

Expanding maps on 2-step infra-nilmanifolds

Abstract A C1-endomorphism f :M→M is expanding if for some Riemannian metric on M there exist c>0, λ>1 such that ∥(Df)mv∥⩾cλm∥v∥ for all v∈TM and all integers m>0. An infra-nilmanifold is the quotient of a connected, simply connected nilpotent Lie group G by a discrete cocompact subgroup π of G⋊C, where C is a compact subgroup of Aut(G). The main result is that every 2-step infra-nilmanifold admits an expanding map.

Expanding maps, Anosov diffeomorphisms and affine structures on infra-nilmanifolds

Abstract In this paper we present a complete description of Lie algebras admitting an expanding and/or a hyperbolic automorphism in terms of R -gradings of these Lie algebras. Using this, we then show that any infra-nilmanifold which admits an expanding map or an Anosov diffeomorphism also has a complete, affinely flat structure.

Applications of group cohomology to space constructions

From a short exact sequence of crossed modules 1 —► K —> H —> H —» 1 and a 2-cocycle (4>,h) € Z2(G;H), a 4-term cohomology exact sequence ff¿6(G; Z) H¡i:h) (G; H, Z) 4 [J{Hl(G; K):^0ut = ¿out} H2b(G; Z) is obtained. Here the first and the last term are the ordinary (=abelian) cohomology groups, and Z is the center of the crossed module H. The second term is shown to be in one-to-one correspondence with certain geometric constructions, called Seifert fiber space construction. Therefore, it follows that, if both the end terms vanish, the geometric construction exists and is unique. 1. Definitions, notations and main results. For any group G and an element a eG, the symbol p(a) denotes the conjugation by a; that is, p(a)(x) = axa-1 for every x eG. The group of all automorphisms of G is denoted by Aut(G); Inn(G) is the group of all inner automorphisms p(a). A crossed module is a 4-tuple (H, p, IT, 4>) where H and II are groups; p: H —» II and

Maximal Holonomy of Almost Bieberbach Groups for Heis5

: n -+ Aut(ii) are group homomorphisms satisfying

Seifert manifolds with fiber spherical space forms

o/) = /¡, /i(x) o p = p o