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Expander graphs in pure and applied mathematics
 Bull. Amer. Math. Soc. (N.S
"... Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more. In recent years they have started to play an increasing role also in pure mathematics: number th ..."
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Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more. In recent years they have started to play an increasing role also in pure mathematics: number theory, group theory, geometry and more. This expository article describes their constructions and various applications in pure and applied mathematics. This paper is based on notes prepared for the Colloquium Lectures at the
The Galois group of random elements of linear groups
, 2012
"... Abstract. Let F be a finitely generated field of characteristic zero and Γ ≤ GLn(F) a finitely generated subgroup. For γ ∈ Γ, let Gal(F(γ)/F) be the Galois group of the splitting field of the characteristic polynomial of γ over F. We show that the structure of Gal(F(γ)/F) has a typical behaviour dep ..."
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Abstract. Let F be a finitely generated field of characteristic zero and Γ ≤ GLn(F) a finitely generated subgroup. For γ ∈ Γ, let Gal(F(γ)/F) be the Galois group of the splitting field of the characteristic polynomial of γ over F. We show that the structure of Gal(F(γ)/F) has a typical behaviour depending on F, and on the geometry of the Zariski closure of Γ (but not on Γ). 1.
Notes on thin matrix groups
"... These notes were prepared for the MSRI hot topics workshop on superstrong approximation (2012). We give a brief overview of the developments in the theory, especially the fundamental expansion theorem. Applications to diophantine problems on orbits of integer matrix groups, the affine sieve, group ..."
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Cited by 5 (0 self)
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These notes were prepared for the MSRI hot topics workshop on superstrong approximation (2012). We give a brief overview of the developments in the theory, especially the fundamental expansion theorem. Applications to diophantine problems on orbits of integer matrix groups, the affine sieve, group theory, gonality of curves and Heegaard genus of hyperbolic three manifolds, are given. We also discuss the ubiquity of thin matrix groups in various contexts, and in particular that of monodromy groups.
SIEVE IN EXPANSION
, 2010
"... This report presents recent works extending sieve methods, from their classical setting, to new situations characterized by the targeting of sets with exponential growth, arising often from discrete groups like SLm(Z) or sufficiently big subgroups. A recent lecture of Sarnak [54] mentions some of th ..."
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This report presents recent works extending sieve methods, from their classical setting, to new situations characterized by the targeting of sets with exponential growth, arising often from discrete groups like SLm(Z) or sufficiently big subgroups. A recent lecture of Sarnak [54] mentions some of the original motivation (related to the
GENERIC ELEMENTS IN ZARISKIDENSE SUBGROUPS AND ISOSPECTRAL LOCALLY SYMMETRIC SPACES
, 2013
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Large Galois groups with applications to Zariski density. arXiv preprint arXiv:1312.3009
, 2013
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SIEVE IN DISCRETE GROUPS, ESPECIALLY SPARSE
"... Abstract. We survey the recent applications and developments of sieve methods related to discrete groups, especially in the case of infinite index subgroups of arithmetic groups. 1. ..."
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Abstract. We survey the recent applications and developments of sieve methods related to discrete groups, especially in the case of infinite index subgroups of arithmetic groups. 1.
The Chebotarev invariant of a finite group
"... Abstract. We consider invariants of a finite group G related to the number of random (independent, uniformly distributed) conjugacy classes which are required to generate it. These invariants are intuitively related to problems of Galois theory. We find grouptheoretic expressions for them and inves ..."
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Abstract. We consider invariants of a finite group G related to the number of random (independent, uniformly distributed) conjugacy classes which are required to generate it. These invariants are intuitively related to problems of Galois theory. We find grouptheoretic expressions for them and investigate their values both theoretically and numerically. 1.
Generic phenomena in groups: some answers and many questions
 Thin groups and superstrong approximation, number 61 in Math. Sci. Res. Inst. Publ
, 2014
"... To the memory of Bill Thurston, with gratitude Abstract. We give a survey of some known results and of the many open questions in the study of generic phenomena in geometrically interesting groups. Contents ..."
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To the memory of Bill Thurston, with gratitude Abstract. We give a survey of some known results and of the many open questions in the study of generic phenomena in geometrically interesting groups. Contents