Finiteness of the fundamental group of compact convex integral polyhedra (Q1316509)

Let \(\Delta_i\), \(i = 1, \ldots, k\), be given compact, convex integral polyhedra in \(\mathbb{R}^n\). Considering the ``combinatorial connectivity'' of \(\{\Delta_1, \ldots, \Delta_k\}\) given by \[ \alpha (\Delta_1, \ldots, \Delta_k) : = \min \left\{ \dim \Bigl( \sum_{i \in I} \Delta_i \Bigr) - |I |; \quad I \subset \{1, \ldots, k\},\;I \neq \emptyset \right\}, \] the author shows that the boundary lattice group (resp. fundamental group) of \(\{\Delta_1, \ldots, \Delta_k\}\) has rank \(n\) if and only if \[ \alpha (\Delta_1, \ldots, \Delta_k) \geq 1. \]