Uniform expansion bounds for Cayley graphs of SL2(Fp)

By Jean Bourgain, Alex Gamburd. 2008

314 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/401cd19b2b3d364d5fdf6a78ac34bed4baa5a6af

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Mathematical Physics

Citations in the corpus

Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds (2008, background): “s obtained in [5]. The diﬀerence between our statements and theirs is that we can count with an archimedean norm. Neither method can specify R1 or R2, due to the lack of an explicit spectral gap, see [3]. APOLLONIAN CIRCLE PACKING 6 (see the discussion in [5, Ex D]). The constants cand c2 are detail”
A Strong Tits Alternative (2008, methodology): “This way of interpreting the result allows to derive, via an effective Nullstellensatz, several corollaries about the girth in finite simple groups of Lie type, as well as some diophantine properties of non-discrete subgroups of GLn(C), in the spirit of the works of Kaloshin-Rodnianski, Helfgott and”
Almost prime Pythagorean triples in thin orbits (2010, background): “) Collecting the results in [3], [6], [5] and the extension from prime to square-free of [12], we have: 95 Kontorovich and Oh, Almost prime Pythagorean triples in thin orbits”
On asymptotic formulae in some sum–product questions (2018, methodology): “So, our another aim is to obtain some new applications (also, see recent papers [32], [31] where other applications were found) of classical graph (group) expansion phenomena, see [9], [21], [18], [22], [44] and others.”
NEW RESULTS ON SUM‐PRODUCT TYPE GROWTH OVER FIELDS (2017, methodology): “ classical graph expansion phenomena appear to be closely related. The former was used in the breakthrough work of Helfgott [20] to prove expansion in SL2(Fp), which then enabled Bourgain and Gamburd [9] and others to change the landscape in the theory of expansion in groups. Conversely, Vinh in [56”
Expansion in $SL_d(O_K/I)$, $I$ square-free (2010, methodology): “We make use of the following result which appeared first implicitly in the proof of Proposition 2 in Bourgain, Gamburd [5].”
Markoff Triples and Strong Approximation (2015, methodology): “It uses the expansion theory (cf. 16]) in SL 2 ( p ) ([4]) as well as the “projective Szemeredi–Trotter Theorem” proved in [3] for pairs of points in ( p ) , which are incident by a subset of 2 p 1 1 | = 2 −”
Sector Estimates for Hyperbolic Isometries (2010, methodology): “Infinite volume spectral gaps are known ([G], [ BG2 ], [BGS]) for prime and squarefree q. The method in [G] applies also for arbitrary composite q, see e.g.”
Expansion in finite simple groups of Lie type (2013, methodology): “We now establish (9), following the arguments of Bourgain and Gamburd [8].”
Random walks in compact groups (2012, background): “The statement of the following theorem is very technical, so we first explain it informally: We suppose that G is a quasirandom profinite group and partition its finite index normal subgroups into two sets: Ω1∪̇Ω2.”