Antonio Montero

An (n+1)-toroid is a quotient of a tessellation of the n-dimensional Euclidean space with a lattice group. Toroids are generalisations of maps on the torus to higher dimensions and also provide examples of abstract polytopes. Equivelar toroids are those that are induced by regular tessellations. In this paper we present a classification of equivelar (n+1)-toroids with at most n flag-orbits; in particular, we discuss a classification of 2-orbit toroids of arbitrary dimension.

Classical geometric and topological operations on polyhedra, maps and polytopes often give rise to structures with the same symmetry group as the original one, but with more flags. In this paper we introduce the notion of voltage operations on maniplexes, as a way to unify the study of such operations and generalize them to other geometric and combinatorial structures such as abstract polytopes, hypermaps, maniplexes or hypertopes. This can be done since our technique provides a way to study classical operations in a graph theoretic setting, and thus to apply a voltage operation one only needs that the combinatorial structure in hand can be understood as an n-valent properly n-edge colored graph. For example, in the case of abstract polytopes, the partial order can be encoded into the so-called flag graph of the polytope and the voltage operation is therefore applied to such flag graph to then be recovered as a partial order. We focus on studying the interactions between voltage operations and the symmetries of the operated object, and show that these operations can be potentially used to build maniplexes with prescribed symmetry type graphs. Moreover, a complete characterization of when an operation can be seen as a voltage operation is given.

Abstract polytopes are combinatorial objects that generalise geometric objects such as convex polytopes, maps on surfaces and tilings of the space. Chiral polytopes are those abstract polytopes that admit full combinatorial rotational symmetry but do not admit reflections. In this paper we build chiral polytopes whose facets (maximal faces) are isomorphic to a prescribed regular cubic tessellation of the n-dimensional torus \(n ≥2\). As a consequence, we prove that for every \(d ≥3\)there exist infinitely many chiral d-polytopes.