The intersection graph of a finite simple group has diameter at most 5

Let G be a non-abelian finite simple group. In addition, let $$\Delta _G$$ Δ G be the intersection graph of G, whose vertices are the proper non-trivial subgroups of G, with distinct subgroups joined by an edge if and only if they intersect non-trivially. We prove that the diameter of $$\Delta _G$$ Δ G has a tight upper bound of 5, thereby resolving a question posed by Shen (Czechoslov Math J 60(4):945–950, 2010). Furthermore, a diameter of 5 is achieved only by the baby monster group and certain unitary groups of odd prime dimension.

Admitting a Coarse Embedding is Not Preserved Under Group Extensions

We construct a finitely generated group which is an extension of two finitely generated groups coarsely embeddable into Hilbert space but which itself does not coarsely embed into Hilbert space. Our construction also provides a new infinite monster group: the first example of a finitely generated group that does not coarsely embed into Hilbert space and yet does not contain a weakly embedded expander.

UNIVERSAL NEAR-HORIZON CONFORMAL STRUCTURE AND BLACK HOLE ENTROPY

It is shown that a massless scalar probe reveals a universal near-horizon conformal structure for a wide class of black holes, including the BTZ. The central charge of the corresponding Virasoro algebra contains information about the black hole. With a suitable quantization condition on the central charge, the CFT associated with the black hole in our approach is consistent with the recent observation of Witten, where the dual theory for the BTZ in the AdS/CFT framework has been identified with the construction of Frenkel, Lepowsky and Meurman. This CFT admits the Fischer–Griess monster group as its symmetry. The logarithm of the dimension of a specific representation of the monster group has been identified by Witten as the entropy of the BTZ black hole. Our algebraic approach shows that a wide class of black holes share the same near-horizon conformal structure as that for the BTZ. With a suitable quantization condition, the CFT's for all these black holes in our formalism can be identified with the FLM model, although not through the AdS/CFT correspondence. The corresponding entropy for the BTZ provides a lower bound for the entropy of this entire class of black holes.

On the Action of the Sporadic Simple Baby Monster Group on its Conjugacy Class 2B

We determine the character table of the endomorphism ring of the permutation module associated with the multiplicity-free action of the sporadic simple Baby Monster group B on its conjugacy class 2B, where the centraliser of a 2B-element is a maximal subgroup of shape 21+22.Co2. This is one of the first applications of a new general computational technique to enumerate big orbits.