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The density of representation degrees

Martin Liebeck;  Dan Segal;  Aner Shalev — 2012

Journal of the European Mathematical Society

For a group G and a positive real number x , define d G ( x ) to be the number of integers less than x which are dimensions of irreducible complex representations of G . We study the asymptotics of d G ( x ) for algebraic groups, arithmetic groups and finitely generated linear groups. In particular we prove an “alternative” for finitely generated linear groups G in characteristic zero, showing that either there exists α > 0 such that d G ( x ) > x α for all large x , or G is virtually abelian (in which case d G ( x ) is bounded).

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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