5,762 research outputs found
Free group automorphisms with many fixed points at infinity
A concrete family of automorphisms alpha_n of the free group F_n is
exhibited, for any n > 2, and the following properties are proved: alpha_n is
irreducible with irreducible powers, has trivial fixed subgroup, and has 2n-1
attractive as well as 2n repelling fixed points at bdry F_n. As a consequence
of a recent result of V Guirardel there can not be more fixed points on bdry
F_n, so that this family provides the answer to a question posed by G Levitt.Comment: This is the version published by Geometry & Topology Monographs on 29
April 200
Kirillov--Reshetikhin crystals for nonexceptional types
We provide combinatorial models for all Kirillov--Reshetikhin crystals of
nonexceptional type, which were recently shown to exist. For types D_n^(1),
B_n^(1), A_{2n-1}^(2) we rely on a previous construction using the Dynkin
diagram automorphism which interchanges nodes 0 and 1. For type C_n^(1) we use
a Dynkin diagram folding and for types A_{2n}^(2), D_{n+1}^(2) a similarity
construction. We also show that for types C_n^(1) and D_{n+1}^(2) the analog of
the Dynkin diagram automorphism exists on the level of crystals.Comment: 35 pages; typos fixed; to appear in Advances in Mathematic
A Survey of Graphs of Minimum Order with Given Automorphism Group
We survey vertex minimal graphs with prescribed automorphism group. Whenever possible, we also investigate the construction of such minimal graphs, confirm minimality, and prove a given graph has the correct automorphism group
Surfaces with given Automorphism Group
Frucht showed that, for any finite group , there exists a cubic graph such
that its automorphism group is isomorphic to . For groups generated by two
elements we simplify his construction to a graph with fewer nodes. In the
general case, we address an oversight in Frucht's construction. We prove the
existence of cycle double covers of the resulting graphs, leading to simplicial
surfaces with given automorphism group. For almost all finite non-abelian
simple groups we give alternative constructions based on graphic regular
representations. In the general cases for , we provide
alternative constructions of simplicial spheres. Furthermore, we embed these
surfaces into the Euclidean 3-Space with equilateral triangles such that the
automorphism group of the surface and the symmetry group of the corresponding
polyhedron in are isomorphic
Recognizing Graph Theoretic Properties with Polynomial Ideals
Many hard combinatorial problems can be modeled by a system of polynomial
equations. N. Alon coined the term polynomial method to describe the use of
nonlinear polynomials when solving combinatorial problems. We continue the
exploration of the polynomial method and show how the algorithmic theory of
polynomial ideals can be used to detect k-colorability, unique Hamiltonicity,
and automorphism rigidity of graphs. Our techniques are diverse and involve
Nullstellensatz certificates, linear algebra over finite fields, Groebner
bases, toric algebra, convex programming, and real algebraic geometry.Comment: 20 pages, 3 figure
The equivariant topology of stable Kneser graphs
The stable Kneser graph , , , introduced by Schrijver
\cite{schrijver}, is a vertex critical graph with chromatic number , its
vertices are certain subsets of a set of cardinality . Bj\"orner and de
Longueville \cite{anders-mark} have shown that its box complex is homotopy
equivalent to a sphere, \Hom(K_2,SG_{n,k})\homot\Sphere^k. The dihedral group
acts canonically on , the group with 2 elements acts
on . We almost determine the -homotopy type of
\Hom(K_2,SG_{n,k}) and use this to prove the following results. The graphs
are homotopy test graphs, i.e. for every graph and such
that \Hom(SG_{2s,4},H) is -connected, the chromatic number
is at least . If and then
is not a homotopy test graph, i.e.\ there are a graph and an such
that \Hom(SG_{n,k}, G) is -connected and .Comment: 34 pp
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