Problems on Bieberbach Groups and Flat Manifolds

Szczepański, Andrzej

doi:10.1007/s10711-006-9056-1

Cited by 11 publications

(8 citation statements)

References 29 publications

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“…However, the group

Out (π_{1} false(M false))

is generally complicated, partly because the holonomy group

normalΦ

is subtle. For some open problems relating

Out (π_{1} false(M false))

, see Szczepański . Our characterization does not use

Out (π_{1} false(M false))

.…”

Section: Introductionmentioning

confidence: 99%

Symmetries of flat manifolds, Jordan property and the general Zimmer program

2019

J. London Math. Soc.

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We obtain a sufficient and necessary condition for a finite group to act effectively on a closed flat manifold. Let G = En(R), EUn(R, Λ), SAut(Fn) or SOut(Fn). As applications, we prove that when n 3 every group action of G on a closed flat manifold M k (k < n) by homeomorphisms is trivial. This confirms a conjecture related to Zimmer's program for flat manifolds. Moreover, it is also proved that the group of homeomorphisms of closed flat manifolds are Jordan with Jordan constants depending only on dimensions.

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“…However, the group

Out (π_{1} false(M false))

is generally complicated, partly because the holonomy group

normalΦ

is subtle. For some open problems relating

Out (π_{1} false(M false))

, see Szczepański . Our characterization does not use

Out (π_{1} false(M false))

.…”

Section: Introductionmentioning

confidence: 99%

Symmetries of flat manifolds, Jordan property and the general Zimmer program

2019

J. London Math. Soc.

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“…Consider an automorphism of G 2 which corresponds to the automorphism of the permutation group defined by (1 2) −→ (1 2)(5 6), (1 2 3 4) −→ (1 2 3 4)(5, 6), (5, 6) −→ (5,6).…”

Section: Crystallographic Groups With Trivial Center and Outer Automomentioning

confidence: 99%

Crystallographic groups with trivial center and outer automorphism group

Lutowski¹,

Szczepański²

2017

Math. Proc. Camb. Phil. Soc.

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“…We refer to [15,20,24] for studies on the realization of flat manifolds as boundary of hyperbolic manifolds. Also, in recent years, the η invariant has been computed in several particular cases (see for instance [4,9,11,13,16,21,22,25,26]).…”

Section: Introductionmentioning

confidence: 99%

The η invariant of the Atiyah–Patodi–Singer operator on compact flat manifolds

Miatello

Podestá

2012

Ann Glob Anal Geom

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Let D be the boundary operator defined by Atiyah, Patodi and Singer, acting on smooth even forms of a compact orientable Riemannian manifold M. In continuation of our previous study, we deal with the problem of computing explicitly the η invariant η = η(M) for any orientable compact flat manifold M. After giving an explicit expression for η(s) in the case of cyclic holonomy group, we obtain a combinatorial formula that reduces the computation to the cyclic case. We illustrate the method by determining η(0) for several infinite families, some of them having non-abelian holonomy groups. For cyclic groups of odd prime order p ≥ 7, η(s) can be expressed as a multiple of L χ (s), an L-function associated to a quadratic character mod p, while η(0) is a (non-zero) integral multiple of the class number h − p of the number field Q( √ − p). In the case of metacyclic groups of odd order pq, with p, q primes, we show that η(0) is a rational multiple of h − p .

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Problems on Bieberbach Groups and Flat Manifolds

Abstract: We present about twenty conjectures, problems and questions about flat manifolds. Many of them build the bridges between the flat world and representation theory of the finite groups, hyperbolic geometry and dynamical systems.

Cited by 11 publications

References 29 publications

Symmetries of flat manifolds, Jordan property and the general Zimmer program

Symmetries of flat manifolds, Jordan property and the general Zimmer program

Crystallographic groups with trivial center and outer automorphism group

The η invariant of the Atiyah–Patodi–Singer operator on compact flat manifolds

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