|
Session 35. Topological fixed point theory and related topics
|
Holonomy groups of flat manifolds with \(R_\infty\) property |
Rafał Lutowski, University of Gdańsk, Poland
|
The talk is based on the joint work with Andrzej Szczepański
|
|
|
Let \(M\) be a closed Riemannian manifold. We say that \(M\) has the \(R_\infty\) property
if for every homeomorphism \(f \colon M \to M\) the Reidemeister number \(R(f)\) is
equal to \(\infty\). \(M\) is flat if, at any point, its sectional curvature is equal to zero.
We investigate a relation between the holonomy representation \(\rho\) of a flat
manifold \(M\) and the \(R_\infty\) property of \(M\). In particular we consider the case where
the holonomy group of \(M\) is solvable. We show that if \(\rho\) has - in a given
sense unique - \(\mathbb{R}\)-irreducible subrepresentation of odd degree, then \(M\) has the
\(R_\infty\) property.
The result is related to Conjecture 4.8 from [1].
|
|
|
References- K. Dekimpe, B. De Rock, P. Penninckx,
The \(R_\infty\) property for infra-nilmanifolds ,
Topol. Methods Nonlinear Anal. 34 (2009), no.2, 353-373
|
|
| Print version |
|
