39 research outputs found
Cayley graphs of order 30p are Hamiltonian
No full textAbstractSuppose G is a finite group, such that |G|=30p, where p is prime. We show that if S is any generating set of G, then there is a Hamiltonian cycle in the corresponding Cayley graph Cay(G;S)
Horospherical limit points of S-arithmetic groups
No full textAbstract. Suppose Γ is an S-arithmetic subgroup of a connected, semisimple algebraic group G over a global field Q (of any characteristic). It is well known that Γ acts by isometries on a certain CAT(0) metric space X S = v∈S X v , where each X v is either a Euclidean building or a Riemannian symmetric space. For a point ξ on the visual boundary of X S , we show there exists a horoball based at ξ that is disjoint from some Γ-orbit in X S if and only if ξ lies on the boundary of a certain type of flat in X S that we call "Q-good." This generalizes a theorem of G. Avramidi and D. W. Morris that characterizes the horospherical limit points for the action of an arithmetic group on its associated symmetric space X
Left-orderable groups and amenability
No full textThis thesis provides a review of left-ordered groups and amenable groups. These are used to investigate a conjecture by Peter Linnell, which relates the existence of a nonabelian free group to a strengthening of left-orderability, by examining three research articles. Lastly, we propose a possible generalisation of a theorem by Dave Witte Morris and Peter Linnell.February 202
2-GENERATED CAYLEY DIGRAPHS ON NILPOTENT GROUPS HAVE HAMILTONIAN PATHS
No full textSuppose G is a nilpotent, finite group. We show that if {a,b} is any 2-element generating set of G, then the corresponding Cayley digraph Cay(G;a,b) has a hamiltonian path. This implies that every connected, cubic Cayley graph on G has a hamiltonian path
ℤ_3^8 is not a CI-group
No full textOpen access article. Creative Commons Attribution 4.0 International license (CC BY 4.0) appliesA Cayley graph Cay(G, S) has the CI (Cayley Isomorphism) property if for every isomorphic graph Cay(G, T), there is a group automorphism α of G such that Sα = T. The DCI (Directed Cayley Isomorphism) property is defined analogously on digraphs. A group G is a CI-group if every Cayley graph on G has the CI property, and is a DCI-group if every Cayley digraph on G has the DCI property. Since a graph is a special type of digraph, this means that every DCI-group is a CI-group, and if a group is not a CI-group then it is not a DCI-group.
In 2009, Spiga showed that ℤ38 is not a DCI-group, by producing a digraph that does not have the DCI property. He also showed that ℤ35 is a DCI-group (and therefore also a CI-group). Until recently the question of whether there are elementary abelian 3-groups that are not CI-groups remained open. In a recent preprint with Dave Witte Morris, we showed that ℤ310 is not a CI-group. In this paper we show that with slight modifications, the underlying undirected graph of order 38 described by Spiga is does not have the CI property, so ℤ38 is not a CI-group.Ye
ON CAYLEY DIGRAPHS THAT DO NOT HAVE HAMILTONIAN PATHS
No full textWe construct an infinite family Cay(Gi; ai, bi) of connected, 2-generated Cayley digraphs that do not have hamiltonian paths, such that the orders of the generators ai and bi are unbounded. We also prove that if G is any finite group with |[G,G] | ≤ 3, then every connected Cayley digraph on G has a hamiltonian path (but the conclusion does not always hold when |[G,G] | = 4 or 5)
Non-left-orderability of lattices in higher-rank semisimple Lie groups (after Deroin and Hurtado)
No full textLet be a connected, semisimple, real Lie group with finite centre, with real rank at least two. B.Deroin and S.Hurtado recently proved the 30-year-old conjecture that no irreducible lattice in has a left-invariant total order. (Equivalently, they proved that no such lattice has a nontrivial, orientation-preserving action on the real line.) We will explain many of the main ideas of the proof, by using them to prove the analogous result for lattices in -adic semisimple groups. The -adic case is easier, because some of the technical issues do not arise.30 page
