We show that every finitely generated non-abelian free group Γ admits a spherically transitive action on a rooted tree T such that the action of Γ on the boundary of T is not essentially free. This reproves a result of Bergeron and Gaboriau. The existence of such an action answers a question of Grigorchuk, Nekrashevich and Sushchanskii.

We study (G, 2)-arc-transitive graphs for innately transitive permutation groups G such that G can be embedded into a wreath product SymΓwr Sl acting in product action on Γ. We find two such connected graphs: the first is Sylvester’s double six graph with 36 vertices, while the second is a graph with 120 vertices whose automorphism group is Aut Sp(4, 4). We prove that under certain conditions n...

A transitive decomposition is a pair ðG;PÞ where G is a graph and P is a partition of the arc set of G, such that there exists a group of automorphisms of G which leaves P invariant and transitively permutes the parts in P. This paper concerns transitive decompositions where the group is a primitive rank 3 group of ‘grid’ type. The graphs G in this case are either products or Cartesian products...

A homeomorphism f of a manifold M is called H1-transitive if there is a transitive lift of an iterate of f to the universal Abelian cover M̃ . Roughly speaking, this means that f has orbits which repeatedly and densely explore all elements of H1(M). For a rel pseudo-Anosov map φ of a compact surface M we show that the following are equivalent: (a) φ is H1-transitive, (b) the action of φ on H1(M)...

A transitive decomposition is a pair (Γ,P) where Γ is a graph and P is a partition of the arc set of Γ such that there is a subgroup of automorphisms of Γ which leaves P invariant and transitively permutes the parts in P. In an earlier paper we gave a characterisation of G-transitive decompositions where Γ is the graph product Km×Km and G is a rank 3 group of product action type. This character...

A finite set X in some Euclidean space Rn is called Ramsey if for any k there is a d such that whenever Rd is k-coloured it contains a monochromatic set congruent to X. This notion was introduced by Erdős, Graham, Montgomery, Rothschild, Spencer and Straus, who asked if a set is Ramsey if and only if it is spherical, meaning that it lies on the surface of a sphere. This question (made into a co...

In this paper, we study modular categories whose Galois group action on their simple objects are transitive. We show that such admit unique factorization into prime transitive factors. The representations of $${\text {SL}}_2({\mathbb {Z}})$$ associated with proven to be minimal and irreducible. Using the Verlinde formula, characterize as conjugates adjoint subcategory quantum category $${\mathc...

By use of Green 's functions f o r the diffusional motion a very concise formulation and computation of correlation functions is possible. For an axially symmetric overall diffusion with internal rotation about one and two axes correlation functions of a second rank spherical tensor are calculated. The results comprise all the solutions of the pertinent problem as given so far and allow the ext...

How do children map linguistic representations onto the conceptual structures that they encode? In the present studies, we provided 3–4-year-old children with minimal-pair scene contrasts in order to determine the effect of particular event properties on novel verb learning. Specifically, we tested whether spatiotemporal cues to causation also inform children’s interpretation of transitive verb...

For two normal edge-transitive Cayley graphs on groups H and K which have no common direct factor and $gcd(|H/H^prime|,|Z(K)|)=1=gcd(|K/K^prime|,|Z(H)|)$, we consider four standard products of them and it is proved that only tensor product of factors can be normal edge-transitive.