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Search: MSC category 05C25 ( Graphs and abstract algebra (groups, rings, fields, etc.) [See also 20F65] )

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1. CJM 2011 (vol 63 pp. 1254)

D'Azevedo, Antonio Breda; Jones, Gareth A.; Schulte, Egon
Constructions of Chiral Polytopes of Small Rank
An abstract polytope of rank $n$ is said to be chiral if its automorphism group has precisely two orbits on the flags, such that adjacent flags belong to distinct orbits. This paper describes a general method for deriving new finite chiral polytopes from old finite chiral polytopes of the same rank. In particular, the technique is used to construct many new examples in ranks $3$, $4$, and $5$.

Keywords:abstract regular polytope, chiral polytope, chiral maps
Categories:51M20, 52B15, 05C25

2. CJM 2009 (vol 61 pp. 1300)

Hubard, Isabel; Orbani\'c, Alen; Weiss, Asia Ivi\'c
Monodromy Groups and Self-Invariance
For every polytope $\mathcal{P}$ there is the universal regular polytope of the same rank as $\mathcal{P}$ corresponding to the Coxeter group $\mathcal{C} =[\infty, \dots, \infty]$. For a given automorphism $d$ of $\mathcal{C}$, using monodromy groups, we construct a combinatorial structure $\mathcal{P}^d$. When $\mathcal{P}^d$ is a polytope isomorphic to $\mathcal{P}$ we say that $\mathcal{P}$ is self-invariant with respect to $d$, or $d$-invariant. We develop algebraic tools for investigating these operations on polytopes, and in particular give a criterion on the existence of a $d$\nobreakdash-auto\-morphism of a given order. As an application, we analyze properties of self-dual edge-transitive polyhedra and polyhedra with two flag-orbits. We investigate properties of medials of such polyhedra. Furthermore, we give an example of a self-dual equivelar polyhedron which contains no polarity (duality of order 2). We also extend the concept of Petrie dual to higher dimensions, and we show how it can be dealt with using self-invariance.

Keywords:maps, abstract polytopes, self-duality, monodromy groups, medials of polyhedra
Categories:51M20, 05C25, 05C10, 05C30, 52B70

3. CJM 2002 (vol 54 pp. 795)

Möller, Rögnvaldur G.
Structure Theory of Totally Disconnected Locally Compact Groups via Graphs and Permutations
Willis's structure theory of totally disconnected locally compact groups is investigated in the context of permutation actions. This leads to new interpretations of the basic concepts in the theory and also to new proofs of the fundamental theorems and to several new results. The treatment of Willis's theory is self-contained and full proofs are given of all the fundamental results.

Keywords:totally disconnected locally compact groups, scale function, permutation groups, groups acting on graphs
Categories:22D05, 20B07, 20B27, 05C25

4. CJM 1999 (vol 51 pp. 1226)

McKay, John
Semi-Affine Coxeter-Dynkin Graphs and $G \subseteq \SU_2(C)$
The semi-affine Coxeter-Dynkin graph is introduced, generalizing both the affine and the finite types.

Categories:20C99, 05C25, 14B05

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