The slab theorem for minimal surfaces in $$mathbb {E}(-1,tau )$$

Unlike (mathbb {R}^{3}), the homogeneous spaces (mathbb {E}(-1,tau )) have a great variety of entire vertical minimal graphs. In this paper we explore conditions which guarantee that a minimal surface in (mathbb {E}(-1,tau )) is such a graph. More specifically, we introduce the definition of a generalized slab in (mathbb {E}(-1,tau )) and prove that a properly immersed minimal surface of finite topology inside such a slab region has multi-graph ends. Moreover, when the surface is embedded, the ends are graphs. When the surface is embedded and simply connected, it is an entire graph.