# Comments on "Discrete Groups, Expanding Graphs and Invariant Measures", by Alexander Lubotzky

@article{FournierFacio2019CommentsO, title={Comments on "Discrete Groups, Expanding Graphs and Invariant Measures", by Alexander Lubotzky}, author={Francesco Fournier-Facio}, journal={arXiv: Group Theory}, year={2019} }

This document is a collection of comments that I wrote down while reading the first four chapters of the book "Discrete Groups, Expanding Graphs and Invariant Measures" by Alexander Lubotzky. Most of them are more detailed versions of proofs. Some imprecisions are pointed out and discussed, and some facts referenced in the book are proven. In the appendix we discuss topics of interest in relation to this book, which are however not necessary for its understanding. The aim of this document…

## References

SHOWING 1-10 OF 26 REFERENCES

Expanders and diffusers

- Mathematics
- 1986

Expander graphs are ingredients for making concentrating, switching, and sorting networks, and are closely related to sparse, doubly-stochastic matrices called diffusers. The best explicit examples…

Discrete groups, expanding graphs and invariant measures

- Mathematics, Computer ScienceProgress in mathematics
- 1994

The Banach-Ruziewicz Problem for n = 2, 3 Ramanujan Graphs is solved and the representation theory of PGL 2 is explained.

Isoperimetric inequalities in simplicial complexes

- Mathematics, Computer ScienceComb.
- 2016

A notion of combinatorial expansion for simplicial complexes of general dimension is defined, and it is proved that similar connections exist between the combinatorsial expansion of a complex, and the spectrum of the high dimensional Laplacian defined by Eckmann.

lambda1, Isoperimetric inequalities for graphs, and superconcentrators

- Mathematics, Computer ScienceJ. Comb. Theory, Ser. B
- 1985

This method uses the second smallest eigenvalue of a certain matrix associated with the graph and it is the discrete version of a method used before for Riemannian manifolds for asymptotic isoperimetric inequalities for families of graphs.

Partial Fractions and Four Classical Theorems of Number Theory

- Mathematics, Computer ScienceAm. Math. Mon.
- 2000

It is seen that all four classical theorems follow from just one partial fractions expansion together with special cases of Jacobi’s triple product identity.

A Theory of Graphs

- Computer Science
- 1993

The theory of graphs has broad and important applications, because so many things can be modeled by graphs, and various puzzles and games are solved easily if a little graph theory is applied.

ON THE NUMBER OF LATIN RECTANGLES

- MathematicsBulletin of the Australian Mathematical Society
- 2010

This thesis primarily investigates the number Rk,n of reduced k X n Latin rectangles. Specifically, we find many congruences that involve Rk,n with the aim of improving our understanding of Rk,n.
In…

Ergodic Theory: with a view towards Number Theory

- Mathematics
- 2010

Motivation.- Ergodicity, Recurrence and Mixing.- Continued Fractions.- Invariant Measures for Continuous Maps.- Conditional Measures and Algebras.- Factors and Joinings.- Furstenberg's Proof of…

The Asymptotic Number of Latin Rectangles

- Mathematics
- 1946

1. Introduction. The problem of enumerating n by k Latin rectangles was solved formally by MacMahon [4] using his operational methods. For k = 3, more explicit solutions have been given in [1], [2],…

Lectures on amenability

- Mathematics
- 2002

0 Paradoxical decompositions 0.1 The Banach-Tarski paradox 0.2 Tarski's theorem 0.3 Notes and comments 1 Amenable, locally comact groups 1.1 Invariant means on locally compact groups 1.2 Hereditary…