Abstract. Let H and K be two finite groups with a properly outer action on the II1 factor M. We prove that the group type inclusions M H ⊂ M ⋊K, studied in detail in [BH], have property T in the sense of [Po6] if and only if the group generated by H and K in the outer automorphism group of M has Kazhdan’s property T [K]. This construction yields irreducible, infinite depth subfactors with small Jones indices and property T standard invariant. If H and K are two finite groups with a properly outer action on the II1 factor M, we can compose the two subfactors M H ⊂ M and M ⊂ M ⋊ K to obtain a new inclusion M H ⊂ M ⋊ K. While the Jones index of M H ⊂ M ⋊ K is finite, being equal to |H | · |K|, in general this inclusion can no longer be obtained as a

We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory.

this paper I will describe in more detail some aspects of the impact of computing on research in pure mathematics, and in particular on the use of specialist software to solve mathematical problems.

A graph Γ is symmetric if its automorphism group acts transitively on the arcs of Γ, and s-regular if its automorphism group acts regularly on the set of s-arcs of Γ. Tutte (1947, 1959) showed that every cubic finite symmetric cubic graph is s-regular for some s ≤ 5. We show that a symmetric cubic graph of girth at most 9 is either 1-regular or 2 ′-regular (following the notation of Djokovic), or belongs to a small family of exceptional graphs. On the other hand, we show that there are infinitely many 3-regular cubic graphs of girth 10, so that the statement for girth at most 9 cannot be improved to cubic graphs of larger girth. Also we give a characterisation of the 1- or 2 ′-regular cubic graphs of girth g ≤ 9, proving that with five exceptions these are closely related with quotients of the triangle group ∆(2,3,g) in each case, or of the group 〈x,y |x 2 = y 3 = [x,y] 4 = 1 〉 in the case g = 8. All the 3-transitive cubic graphs and exceptional 1- and 2-regular cubic graphs of girth at most 9 appear in the list of cubic symmetric graphs up to 768 vertices produced by Conder and Dobcsányi (2002); the largest is the 3-regular graph F570 of order 570 (and girth 9). The proofs of the main results are computer-assisted. Keywords: Arc-transitive graph, s-regular graph, girth, triangle group, regular map

Abstract not found