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Threerelator quotients of the modular group
 10 5 , Comm. Algebra 10
, 1987
"... joint work with ..."
Examples of subfactors with property T standard invariant
 Geom. Funct. Anal
, 1999
"... 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 Kaz ..."
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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
Computers and Discovery in Algebraic Graph Theory
 Edinburgh, 2001), Linear Algebra Appl
, 2001
"... We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory. ..."
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We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory.
SemiAutomated Theorem Proving. The Impact Of Computers on Research in Pure Mathematics
 First Asian Technology Conference in Mathematics (Singapore
, 1995
"... 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. ..."
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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.
Symmetric cubic graphs of small girth
"... A graph Γ is symmetric if its automorphism group acts transitively on the arcs of Γ, and sregular if its automorphism group acts regularly on the set of sarcs of Γ. Tutte (1947, 1959) showed that every cubic finite symmetric cubic graph is sregular for some s ≤ 5. We show that a symmetric cubic g ..."
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A graph Γ is symmetric if its automorphism group acts transitively on the arcs of Γ, and sregular if its automorphism group acts regularly on the set of sarcs of Γ. Tutte (1947, 1959) showed that every cubic finite symmetric cubic graph is sregular for some s ≤ 5. We show that a symmetric cubic graph of girth at most 9 is either 1regular 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 3regular 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 3transitive cubic graphs and exceptional 1 and 2regular 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 3regular graph F570 of order 570 (and girth 9). The proofs of the main results are computerassisted. Keywords: Arctransitive graph, sregular graph, girth, triangle group, regular map
A NOTE ON GROUPS ASSOCIATED WITH 4ARCTRANSITIVE CUBIC GRAPHS
"... A cubic (trivalent) graph F is said to be 4arctransitive if its automorphism group acts transitively on the 4arcs of r (where a 4arc is a sequence «;0, vv...,vi of vertices of F such that t;,_j is adjacent to vt for 1 ^ I < 4, and vt1 ^ vi+1 for 1 < i < 4). In his investigations into graphs of ..."
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A cubic (trivalent) graph F is said to be 4arctransitive if its automorphism group acts transitively on the 4arcs of r (where a 4arc is a sequence «;0, vv...,vi of vertices of F such that t;,_j is adjacent to vt for 1 ^ I < 4, and vt1 ^ vi+1 for 1 < i < 4). In his investigations into graphs of this sort, Biggs defined a family of groups 4 + (a m), for m = 3,4,5..., each presented in terms of generators and relations under the additional assumption that the vertices of a circuit of length m are cyclically permuted by some automorphism. In this paper it is shown that whenever m is a proper multiple of 6, the group 4 + (a m) is infinite. The proof is obtained by constructing transitive permutation representations of arbitrarily large degree. 1.