2,292 research outputs found
Regular embeddings of complete bipartite maps: classification and enumeration
No full textThe regular embeddings of complete bipartite graphs Kn, n in orientable surfaces are classified and enumerated, and their automorphism groups and combinatorial properties are determined. The method depends on earlier classifications in the cases where n is a prime power, obtained in collaboration with Du, Kwak, Nedela and koviera, together with results of Itô, Hall, Huppert and Wielandt on factorisable groups and on finite solvable groups. <br/
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
Maps admitting trialities but not dualities
No full textWe use group theory to construct infinite families of maps on surfaces which are invariant under Wilson's map operations of order 3 but not under the operations of order 2, such as duality and Petrie duality
Exotic behaviour of infinite hypermaps
No full textThis is a survey of infinite hypermaps, and of how they can be constructed by using examples and techniques from combinatorial group theory, with particular emphasis on phenomena which have no analogues for finite hypermaps.<br/
Totally chiral maps and hypermaps of small genus
No full textAn orientably regular hypermap is totally chiral if it and its mirror image have no non-trivial common quotients. We classify the totally chiral hypermaps of genus up to 1001, and prove that the least genus of any totally chiral hypermap is 211, attained by twelve orientably regular hypermaps with monodromy group A7 and type (3,4,4) (up to triality). The least genus of any totally chiral map is 631, attained by a chiral pair of orientably regular maps of type {11,4}, together with their duals; their monodromy group is the Mathieu group M11. This is also the least genus of any totally chiral hypermap with non-simple monodromy group, in this case the perfect triple covering 3.A7 of A7. The least genus of any totally chiral map with non-simple monodromy group is 1457, attained by 48 maps with monodromy group isomorphic to the central extension 2.Sz(8).<br/
Cell motility assays
No full textThis report summarises practical aspects to measuring cell motility in culture. The methods described here were discussed at a 1-day European Tissue Culture Society (ETCS-UK) workshop organised by John Masters and Gareth E Jones that was held at University College London on 19th April 2007.This report summarises practical aspects to measuring cell motility in culture. The methods described here were discussed at a 1-day European Tissue Culture Society (ETCS-UK) workshop organised by John Masters and Gareth E Jones that was held at University College London on 19th April 2007
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
- …
