Expansion in SL 2$${(\mathbb{R})}$$ and monotone expanders

This work presents an explicit construction of a family of monotone expanders, which are bi-partite expander graphs whose edge-set is defined by (partial) monotone functions. The family is (roughly) defined by the Möbius action of SL2$${\mathbb{R}}$$ on the interval [0,1]. A key part of the proof is a product-growth theorem for certain subsets of SL2$${\mathbb{R}}$$ . This extends recent results on finite/compact groups to the non-compact scenario. No other proof-of-existence for monotone expanders is known.

[1]  P. Ungar.  A Theorem on Planar Graphs , 1951 .

[2]  Harry Kesten,et al.  Symmetric random walks on groups , 1959 .

[3]  Z. Galil,et al.  Explicit constructions of linear size superconcentrators , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[4]  Maria M. Klawe,et al.  Limitations on Explicit Constructions of Expanding Graphs , 1984, SIAM J. Comput..

[5]  Peter Sarnak,et al.  Bounds for multiplicities of automorphic representations , 1991 .

[6]  D. Spielman,et al.  Expander codes , 1996 .

[7]  K. Simon,et al.  Invariant measures for parabolic IFS with overlaps and random continued fractions , 2001 .

[8]  K. Simon,et al.  Hausdorff dimension of limit sets for parabolic ifs with overlaps , 2001 .

[9]  A. Wigderson,et al.  ENTROPY WAVES, THE ZIG-ZAG GRAPH PRODUCT, AND NEW CONSTANT-DEGREE , 2004, math/0406038.

[10]  J. Bourgain.  On the Erdős-Volkmann and Katz-Tao ring conjectures , 2003 .

[11]  N. Linial,et al.  Expander Graphs and their Applications , 2006 .

[12]  Michael A. Nielsen,et al.  The Solovay-Kitaev algorithm , 2006, Quantum Inf. Comput..

[13]  Jean Bourgain,et al.  On the spectral gap for finitely-generated subgroups of SU(2) , 2007 .

[14]  Zeev Dvir,et al.  Towards Dimension Expanders over Finite Fields , 2008, 2008 23rd Annual IEEE Conference on Computational Complexity.

[15]  J. Bourgain,et al.  Uniform expansion bounds for Cayley graphs of SL2(Fp) , 2008 .

[16]  Terence Tao.  Product set estimates for non-commutative groups , 2008, Comb..

[17]  W. T. Gowers,et al.  Quasirandom Groups , 2007, Combinatorics, Probability and Computing.

[18]  A. Lubotzky,et al.  Dimension expanders , 2008, 0804.0481.

[19]  Venkatesan Guruswami,et al.  Euclidean Sections of with Sublinear Randomness and Error-Correction over the Reals , 2008, APPROX-RANDOM.

[21]  H. Helfgott.  Growth and generation in $\mathrm{SL}_2(\mathbb{Z}/p \mathbb{Z})$ , 2008 .

[22]  V. Kaimanovich,et al.  Matrix random products with singular harmonic measure , 2008, 0807.1015.

[23]  J. Bourgain.  Expanders and dimensional expansion , 2009 .

[24]  Avi Wigderson,et al.  Monotone Expanders: Constructions and Applications , 2010, Theory Comput..

[25]  On the Furstenberg measure and density of states for the Anderson-Bernoulli model at small disorder , 2011, 1101.2148.

[26]  Finitely Supported Measures on S{L}_{2}(\mathbb{R}) Which are Absolutely Continuous at Infinity , 2012 .