Estimativa para o volume de ciclos em variedades hiperbólicas com bordo

Abstract

In this work, we prove two results for the model (M, ∂M) = (Σ, ∂Σ) × S 1, where (Σ, ∂Σ) is a hyperbolic manifold with totally geodesic boundary. In the first result, using the Larry Guth’s techniques and under an upper bound on the volumes of unit balls in the universal cover of (M, ∂M), we show that the volume of a hyperbolic manifold with totally geodesic boundary has a relation to ship its simplicial volume. Moreover, under those hypothesis, we generalize a Hannah Alpert and Kei Funano result, showing that the area of Z2-relative homology class [(Σ, ∂Σ)] in (M, ∂M) is proportional to the area of (Σ, ∂Σ).