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Journal articleLiebeck MW, Praeger CE, Saxl J, 2019,
The classification of 3/2transitive permutation groups and 1/2transitive linear groups
, Proceedings of the American Mathematical Society, Vol: 147, Pages: 50235037, ISSN: 10886826A linear group G ≤ GL(V ), where V is a finite vector space, is called 12transitive if all the Gorbits on the set of nonzero vectors have the same size. We complete the classification of all the 12transitive linear groups. As a consequence we complete the determination of the finite 32transitive permutation groups – the transitive groups for which a pointstabilizerhas all its nontrivial orbits of the same size. We also determine the (k +12)transitive groups for integers k ≥ 2.

Journal articleLiebeck MW, Schul G, Shalev A, 2017,
Rapid growth in finite simple groups
, Transactions of the American Mathematical Society, Vol: 369, Pages: 27658779, ISSN: 10886850We show that small normal subsets A of finite simple groups growvery rapidly – namely, A2 ≥ A2− , where > 0 is arbitrarily small.Extensions, consequences, and a rapid growth result for simple algebraicgroups are also given.

Journal articleLiebeck MW, 2017,
Character ratios for finite groups of Lie type, and applications
, Contemporary Mathematics, Vol: 694, ISSN: 02714132For a nite groupG, acharacter ratiois a complex number of the form (x) (1),wherex2Gand is an irreducible character ofG. Upper bounds for absolutevalues of character ratios, particularly for simple groups, have long been of interest,for various reasons; these include applications to covering numbers, mixing timesof random walks, and the study of word maps. In this article we shall survey someresults on character ratios for nite groups of Lie type, and their applications.Character ratios for alternating and symmetric groups have been studied in greatdepth also { see for example [32, 33] { culminating in the de nitive results andapplications to be found in [20]; but we shall not discuss these here.It is not hard to see the connections between character ratios and group structure. Here are three well known, elementary results illustrating these connections.The rst two go back to Frobenius. Denote by Irr(G) the set of irreducible characters ofG.

Journal articleGonshaw S, Liebeck MW, O'Brien E, 2016,
Unipotent class representatives for finite classical groups
, Journal of Group Theory, Vol: 20, Pages: 505525, ISSN: 14354446We describe explicitly representatives of the conjugacy classes ofunipotent elements of the finite classical groups.

Journal articleSchedler TJ, Proudfoot NJ, 2016,
Poisson–de Rham homology of hypertoric varieties and nilpotent cones
, Selecta Mathematica, Vol: 23, Pages: 179202, ISSN: 10221824We prove a conjecture of Etingof and the second author for hypertoric varieties that the Poisson–de Rham homology of a unimodular hypertoric cone is isomorphic to the de Rham cohomology of its hypertoric resolution. More generally, we prove that this conjecture holds for an arbitrary conical variety admitting a symplectic resolution if and only if it holds in degree zero for all normal slices to symplectic leaves. The Poisson–de Rham homology of a Poisson cone inherits a second grading. In the hypertoric case, we compute the resulting 2variable Poisson–de Rham–Poincaré polynomial and prove that it is equal to a specialization of an enrichment of the Tutte polynomial of a matroid that was introduced by Denham (J Algebra 242(1):160–175, 2001). We also compute this polynomial for S3varieties of type A in terms of Kostka polynomials, modulo a previous conjecture of the first author, and we give a conjectural answer for nilpotent cones in arbitrary type, which we prove in rank less than or equal to 2.

Journal articleGinzburg V, Schedler TJ, 2016,
A new construction of cyclic homology
, Proceedings of the London Mathematical Society, Vol: 112, Pages: 549587, ISSN: 00246115Based on the ideas of Cuntz and Quillen, we give a simple construction of cyclic homology of unital algebras in terms of the noncommutative de Rham complex and a certain differential similar to the equivariant de Rham differential. We describe the Connes exact sequence in this setting. We define equivariant Deligne cohomology and construct, for each 𝑛⩾1 , a natural map from cyclic homology of an algebra to the GL𝑛 ‐equivariant Deligne cohomology of the variety of 𝑛 ‐dimensional representations of that algebra. The bridge between cyclic homology and equivariant Deligne cohomology is provided by extended cyclic homology, which we define and compute here, based on the extended noncommutative de Rham complex introduced previously by the authors.

Journal articleEvans DM, Ghadernezhad Z, Tent K, 2016,
Simplicity of the automorphism groups of some Hrushovski constructions
, Annals of Pure and Applied Logic, Vol: 167, Pages: 2248, ISSN: 18732461We show that the automorphism groups of certain countable structures obtained using the Hrushovski amalgamation method are simple groups. The structures we consider are the ‘uncollapsed’ structures of infinite Morley rank obtained by the ab initio construction and the (unstable) ℵ0categorical pseudoplanes. The simplicity of the automorphism groups of these follows from results which generalize work of Lascar and of Tent and Ziegler.

Journal articleEvans DM, Tsankov T, 2016,
Free actions of free groups on countable structures and property (T)
, Fundamenta Mathematicae, Vol: 232, Pages: 4963, ISSN: 17306329We show that if G is a nonarchimedean, Roelcke precompact Polish group, then G has Kazhdan's property (T). Moreover, if G has a smallest open subgroup of finite index, then G has a finite Kazhdan set. Examples of such G include automorphism groups of countable ωcategorical structures, that is, the closed, oligomorphic permutation groups on a countable set. The proof uses work of the second author on the unitary representations of such groups, together with a separation result for infinite permutation groups. The latter allows the construction of a nonabelian free subgroup of G acting freely in all infinite transitive permutation representations of G.

Journal articleLiebeck MW, Seitz GM, Testerman DM, 2015,
Distinguished unipotent elements and multiplicityfree subgroups of simple algebraic groups
, Pacific Journal of Mathematics, Vol: 279, Pages: 357382, ISSN: 00308730For G a simple algebraic group over an algebraically closed field of characteristic 0, we determine the irreducible representations ρ:G→I(V), where I(V) denotes one of the classical groups SL(V), Sp(V), SO(V), such that ρ sends some distinguished unipotent element of G to a distinguished element of I(V). We also settle a base case of the general problem of determining when the restriction of ρ to a simple subgroup of G is multiplicityfree.

Journal articleLiebeck MW, O'Brien EA, 2015,
Recognition of finite exceptional groups of Lie type
, Transactions of the American Mathematical Society, Vol: 368, Pages: 61896226, ISSN: 10886850Let q be a prime power and let G be an absolutely irreducible subgroup ofGLd(F), where F is a finite field of the same characteristic as Fq, the field of q elements. Assume that G ∼= G(q), a quasisimple group of exceptional Lie type over Fq which is neither a Suzuki nor a Ree group. We present a Las Vegas algorithm that constructs an isomorphism from G to the standard copy of G(q). If G 6∼=3D4(q) with q even, then the algorithm runs in polynomial time, subject to the existence of a discrete log oracle.
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