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A problem of Kollár and Larsen on finite linear groups and crepant resolutions

Robert Guralnick;  Pham Tiep — 2012

Journal of the European Mathematical Society

The notion of age of elements of complex linear groups was introduced by M. Reid and is of importance in algebraic geometry, in particular in the study of crepant resolutions and of quotients of Calabi–Yau varieties. In this paper, we solve a problem raised by J. Kollár and M. Larsen on the structure of finite irreducible linear groups generated by elements of age ≤ 1 . More generally, we bound the dimension of finite irreducible linear groups generated by elements of bounded deviation. As a consequence...

The Ore conjecture

Martin Liebeck;  E.A. O’Brien;  Aner Shalev;  Pham Tiep — 2010

Journal of the European Mathematical Society

The Ore conjecture, posed in 1951, states that every element of every finite non-abelian simple group is a commutator. Despite considerable effort, it remains open for various infinite families of simple groups. In this paper we develop new strategies, combining character-theoretic methods with other ingredients, and use them to establish the conjecture.

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