Tits geometry of cocompact real-analytic Hadamard manifolds of dimension 4

Tits Alternative for Closed Real Analytic 4-manifolds of Nonpositive Curvature

We study subgroups of fundamental groups of real analytic closed 4-manifolds with nonpositive sectional curvature. In particular, we are interested in the following question: if a subgroup of the fundamental group is not virtually free abelian, does it contain a free group of rank two ? The technique involves the theory of general metric spaces of nonpositive curvature.

Bruhat-tits Buildings and Analytic Geometry

This paper provides an overview of the theory of Bruhat-Tits buildings. Besides, we explain how Bruhat-Tits buildings can be realized inside Berkovich spaces. In this way, Berkovich analytic geometry can be used to compactify buildings. We discuss in detail the example of the special linear group. Moreover, we give an intrinsic description of Bruhat-Tits buildings in the framework of non-Archim...

Real-analytic, volume-preserving actions of lattices on 4-manifolds

We prove that any real-analytic, volume-preserving action of a lattice Γ in a simple Lie group with Q-rank(Γ) ≥ 7 on a closed 4-manifold of nonzero Euler characteristic factors through a finite group action.

The Geometry of 4-Manifolds

Fractal Dimension of Graphs of Typical Continuous Functions on Manifolds

If M is a compact Riemannian manifold then we show that for typical continuous function defined on M, the upper box dimension of  graph(f) is as big as possible and the lower box dimension of graph(f) is as small as possible.