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1. CJM 2011 (vol 63 pp. 1254)
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)
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)
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)
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 |