Finite subgroups of algebraic groups

By Michael Larsen, R. Pink. 2011

146 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6498bc9ff5f214c9027e1c7ec1d9c90266f639a5

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

On the diameter of permutation groups (2011, result): “[BGH+] proved that if a set of generators of Sym(n) or Alt(n) contains an element of support size at most 0.63n then the diameter of the Cayley graph is bounded by a polynomial of n. 4Cf. the role of [LP11] (esp. Thm. 4.2, Thm. 6.2), which, in order to provide alternatives to the Classiﬁcation of Fi”
Growth in groups: ideas and perspectives (2013, unknown): “the ﬁrst version was made public in 2008 – and the author had conversed at length with one of the authors of [BGT11] about the ideas involved and the diﬃculties remaining. The preprint of Larsen-Pink [LP11] was available, as were works by Hrushovski-Wagner [HW08] and Hrushovski [Hru12]18. The aim of”
On the growth of $L^2$-invariants for sequences of lattices in Lie groups (2012, background): “By the Larsen–Pink theorem [76], there is a function f : N → N such that a subgroup of GL(n,Fpm) contains a subgroup of index at most f(n) whose non-abelian composition factors are from Lie(p).”
Strongly dense free subgroups of semisimple algebraic groups (2010, background): “However, the proof in [28] for classical groups of bounded rank is easily modified to avoid this (using the weaker result in [32] that there are only finitely many sporadic simple groups that have linear representations of fixed dimension – this does not depend upon CFSG).”
Approximate Subgroups of Linear Groups (2010, background): “Nevertheless a fair amount of what we do in this paper, all of §4 and Appendix A, and in particular the key Larsen-Pink inequality, hold for arbitrary algebraic groups including for instance abelian varieties.”
Expansion in finite simple groups of Lie type (2013, background): “The first step in doing so is a classification [3, 42, 45, 46, 62] of the maximal proper subgroups H of the bounded cover G̃ of a finite simple group G of Lie type, which among other things asserts that such subgroups either live in a proper Zariski-closed subgroup of the mother group G (with bounds”
Growth in finite simple groups of Lie type of bounded rank (2010, methodology): “p, where the bound f(n) depends on n. It was proved by Collins [17] that for n≥ 71 one can take f(n) = (n+ 2)!. Remarkably a (non-eﬀective) version of the above result was obtained by Larsen and Pink [43]without relying on the classiﬁcation of ﬁnite simple groups. It is clear that His a soluble by L”
Notes on Thin Matrix Groups (2012, background): “The general form of this phenomenon for arithmetic groups is known as strong approximation and it is well understood [Platonov and Rapinchuk 1994].”
On the automorphy of l-adic Galois representations with small residual image With an appendix by Robert Guralnick, Florian Herzig, Richard Taylor and Jack Thorne (2011, background): “Without it, the proof still goes through for l sufficiently large (depending on d and ineffective) by appealing to [26] instead of [18].”
Stable group theory and approximate subgroups (2009, methodology): “up theoretic setting, but it does hold for suﬃciently dense subgroups of simple linear groups. The proof uses an idea originating in the Larsen-Pink classiﬁcation of large ﬁnite simple linear groups, [35], somewhat generalized and formulated as a dimension-comparison lemma in [24]. We obtain: Theore”