1,722,416 research outputs found
Edge-transitive embeddings of complete graphs
No full textBuilding on earlier work of Biggs, James, Wilson and the author and on the Graver-Watkins description of the 14 classes of edge-transitive maps, we complete the classification of the edge-transitive embeddings of complete graphs, including those with non-empty boundary.</p
Groups of Automorphisms of Riemann Surfaces and Maps of Genus p+1 Where p is Prime
No full textWe classify compact Riemann surfaces of genus g, where g−1 is a prime p, which have a group of automorphisms of order ρ(g−1)for some integer ρ≥1, and determine isogeny decompositions of the corresponding Jacobian varieties. This extends results of Belolipetzky and the second author for ρ>6, and of the first and third authors for ρ= 3, 4, 5 and 6. As a corollary we classify the orientably regular hypermaps (including maps) of genus p+1, together with the non-orientable regular hypermaps of characteristic −p, with automorphism group of order divisible by the prime p; this extends results of Conder, Širáň and Tucker for maps
Realisation of groups as automorphism groups in permutational categories
No full textIt is shown that in various categories, including many consisting of maps or hypermaps, oriented or unoriented, of a given hyperbolic type, or of coverings of a suitable topological space, every countable group A is isomorphic to the automorphism group of uncountably many non-isomorphic objects, infinitely many of them finite if A is finite. In particular, the latter applies to dessins d’enfants, regarded as finite oriented hypermaps.</p
Infinite Paley graphs
No full textInfinite analogues of the Paley graphs are constructed, based on uncountably many locally finite fields. By using character sum estimates due to Weil, they are shown to be isomorphic to the countable random graph of Erdős, Rényi and Rado
Finite simple automorphism groups of edge-transitive maps
No full textBuilding on earlier results for regular maps and for orientably regular chiral maps, we classify the non-abelian finite simple groups arising as automorphism groups of maps in each of the 14 Graver–Watkins classes of edge-transitive maps.</p
Maximal subgroups of the modular and other groups
No full textIn 1933 B. H. Neumann constructed uncountably many subgroups of SL 2 (Z) {{\rm SL}-{2}(\mathbb{Z})} which act regularly on the primitive elements of Z 2 {\mathbb{Z}^{2}}. As pointed out by Magnus, their images in the modular group PSL 2 (Z) ≅ C 3 ∗ C 2 {{\rm PSL}-{2}(\mathbb{Z})\cong C-{3}∗C-{2}} are maximal nonparabolic subgroups, that is, maximal with respect to containing no parabolic elements. We strengthen and extend this result by giving a simple construction using planar maps to show that for all integers p ≥ 3 {p\geq 3}, q ≥ 2 {q\geq 2} the triangle group Γ = Δ (p, q, ∞) ≅ C p ∗ C q {\Gamma=\Delta(p,q,\infty)\cong C-{p}∗C-{q}} has uncountably many conjugacy classes of nonparabolic maximal subgroups. We also extend results of Tretkoff and of Brenner and Lyndon for the modular group by constructing uncountably many conjugacy classes of such subgroups of Γ which do not arise from Neumann's original method. These maximal subgroups are all generated by elliptic elements, of finite order, but a similar construction yields uncountably many conjugacy classes of torsion-free maximal subgroups of the Hecke groups C p ∗ C 2 {C-{p}∗C-{2}} for odd p ≥ 3 {p\geq 3}. Finally, an adaptation of work of Conder yields uncountably many conjugacy classes of maximal subgroups of Δ (2, 3, r) {\Delta(2,3,r)} for all r ≥ 7 {r\geq 7}.</p
r-regular families of graph automorphisms
No full textAn r-regular family F of permutations on a set V contains, for each pair of vertices u,v∈V, exactly r permutations φ mapping u to v. Earlier, 1-regular families of graph automorphisms were used by Gauyacq to define the quasi-Cayley graphs, a class of vertex-transitive graphs that properly contains the class of Cayley graphs, sharing many of their characteristics, and is properly contained in the class of vertex-transitive graphs. We introduce r-regular families to measure how far a vertex-transitive graph is from being quasi-Cayley. As any automorphism group of a graph Γ=(V,E) acting transitively on V with vertex-stabilizers of order r forms an r-regular family on V, every vertex-transitive graph admits an r-regular family of automorphisms for some r≥1. In general, the smallest r for which such a family exists (which we call the quasi-Cayley deficiency of the graph) might be smaller than the order of the vertex-stabilizer of a smallest vertex-transitive automorphism group of the graph (which we call the Cayley deficiency). We investigate the relations between these two parameters for the class of merged Johnson graphs. We prove the existence of Johnson graphs with arbitrarily large quasi-Cayley deficiency, as well as Johnson graphs for which the difference between their Cayley deficiency and their quasi-Cayley deficiency is arbitrarily large.</p
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