This paper uses combinatorial group theory to help answer some long-standing questions about the genera of orientable surfaces that carry particular kinds of regular maps. By classifying all orientably-regular maps whose automorphism group has order coprime to , where is the genus, all orientably-regular maps of genus for prime are determined. As a consequence, it is shown that orientable surfaces of infinitely many genera carry no regular map that is chiral (irreflexible), and that orientable...
The Mordell–Lang conjecture describes the intersection of a finitely generated subgroup with a closed subvariety of a semiabelian variety. Equivalently, this conjecture describes the intersection of closed subvarieties with the set of images of the origin under a finitely generated semigroup of translations. We study the analogous question in which the translations are replaced by algebraic group endomorphisms (and the origin is replaced by another point). We show that the conclusion of the Mordell–Lang...
The distinguishing number D(G) of a graph G is the minimum number of colors needed to color the vertices of G such that the coloring is preserved only by the trivial automorphism. In this paper we improve results about the distinguishing number of Cartesian products of finite and infinite graphs by removing restrictions to prime or relatively prime factors.
Download Results (CSV)