Author | Lubotzky, Alexander. author |
---|---|

Title | Discrete Groups, Expanding Graphs and Invariant Measures [electronic resource] / by Alexander Lubotzky |

Imprint | Basel : Birkhรคuser Basel : Imprint: Birkhรคuser, 1994 |

Connect to | http://dx.doi.org/10.1007/978-3-0346-0332-4 |

Descript | XI, 196 p. online resource |

SUMMARY

In the last ?fteen years two seemingly unrelated problems, one in computer science and the other in measure theory, were solved by amazingly similar techniques from representation theory and from analytic number theory. One problem is the - plicit construction of expanding graphs (ยซexpandersยป). These are highly connected sparse graphs whose existence can be easily demonstrated but whose explicit c- struction turns out to be a dif?cult task. Since expanders serve as basic building blocks for various distributed networks, an explicit construction is highly des- able. The other problem is one posed by Ruziewicz about seventy years ago and studied by Banach [Ba]. It asks whether the Lebesgue measure is the only ?nitely additive measure of total measure one, de?ned on the Lebesgue subsets of the n-dimensional sphere and invariant under all rotations. The two problems seem, at ?rst glance, totally unrelated. It is therefore so- what surprising that both problems were solved using similar methods: initially, Kazhdan's property (T) from representation theory of semi-simple Lie groups was applied in both cases to achieve partial results, and later on, both problems were solved using the (proved) Ramanujan conjecture from the theory of automorphic forms. The fact that representation theory and automorphic forms have anything to do with these problems is a surprise and a hint as well that the two questions are strongly related

CONTENT

Expanding Graphs -- The Banach-Ruziewicz Problem -- Kazhdan Property (T) and its Applications -- The Laplacian and its Eigenvalues -- The Representation Theory of PGL 2 -- Spectral Decomposition of L 2(G(?)\G(A)) -- Banach-Ruziewicz Problem for n = 2, 3; Ramanujan Graphs -- Some More Discrete Mathematics -- Distributing Points on the Sphere -- Open Problems

Mathematics
Group theory
Functions of real variables
Differential geometry
Number theory
Mathematics
Group Theory and Generalizations
Real Functions
Number Theory
Differential Geometry