By Alex Lubotzky, Jonathan D. Rogawski

ISBN-10: 3034603312

ISBN-13: 9783034603317

ISBN-10: 376435075X

ISBN-13: 9783764350758

The ebook offers the ideas to 2 difficulties: the 1st is the development of increasing graphs – graphs that are of primary value for verbal exchange networks and computing device technology; the second one is the Ruziewicz challenge about the finitely additive invariant measures on spheres. either difficulties have been partly solved utilizing the Kazhdan estate (T) from illustration thought of semi-simple Lie teams. Later, whole soultions have been received for either difficulties utilizing the Ramanujan conjecture from analytic quantity conception. the writer, who performed a big position in those advancements, explains the 2 difficulties and their ideas from a standpoint which unearths why these types of doubtless unrelated themes are so interconnected. The unified strategy exhibits interrelations among varied branches of arithmetic corresponding to graph thought, degree idea, Riemannian geometry, discrete subgroups of Lie teams, illustration idea and analytic quantity thought. distinct efforts have been made to make the publication available to graduate scholars in arithmetic and computing device technology. a couple of difficulties and proposals for additional examine are awarded. stories: "This interesting booklet marks the genesis of a brand new box. it's a box during which one passes from side to side at will throughout the taking a look glass dividing the discrete from the continual. (...) The ebook is an enthralling mix of issues from team conception (finite and infinite), combinatorics, quantity conception, harmonic analysis." - Zentralblatt MATH "The Appendix, written via J. Rogawski, explains the Jacquet-Langlands concept and shows Deligne’s facts of the Petersson-Ramanujan conjecture. it's going to advantage its personal evaluation. (...) In end, this can be a incredible manner of transmitting contemporary mathematical study without delay "from the manufacturer to the consumer." - MathSciNet "The e-book is available to mature graduate scholars in arithmetic and theoretical computing device technology. it's a great presentation of a gem on the border of research, geometry, algebra and combinatorics. those that take the hassle to look what occurs at the back of the scene won’t remorse it." - Acta Scientiarum Mathematicarum

Show description

Read Online or Download Discrete Groups, Expanding Graphs and Invariant Measures PDF

Similar differential geometry books

Norman Steenrod's The topology of fibre bundles PDF

Fibre bundles, now a vital part of differential geometry, also are of serious significance in sleek physics - similar to in gauge conception. This e-book, a succinct creation to the topic by means of renown mathematician Norman Steenrod, was once the 1st to provide the topic systematically. It starts off with a basic advent to bundles, together with such themes as differentiable manifolds and protecting areas.

Read e-book online Differential geometry and complex analysis: a volume PDF

Chavel I. , Farkas H. M. (eds. ) Differential geometry and intricate research (Springer, 1985)(ISBN 354013543X)(236s)

Get Theorems on regularity and singularity of energy minimizing PDF

The purpose of those lecture notes is to offer an basically self-contained advent to the fundamental regularity conception for power minimizing maps, together with contemporary advancements about the constitution of the singular set and asymptotics on method of the singular set. really expert wisdom in partial differential equations or the geometric calculus of adaptations is no longer required.

Additional info for Discrete Groups, Expanding Graphs and Invariant Measures

Sample text

So, if the image of Γ in SL n ( k ν ) is precompact for every ν, the image of Γ in SL n (‫ ށ‬k ) is precompact. So it is precompact and discrete. Hence, Γ is finite. ᮀ The following proposition is due to Watatani [Wt] (see also Alperin [Alp] and more generally Adams and Spatzier [AS]). We give here a different proof. 10. Let Γ be a group with property (T) acting on a locally finite tree X without inversions. Then Γ fixes a vertex of X. Proof. Let Y be a minimal Γ-invariant subtree of X. Then we can assume that Γ ⊆ Aut(Y).

3) there is a finite subset A ⊆ Γ such that |A sA| < ε|A| for every s ∈ S ∪ S −1 . For N ∈ ᏸ, set a function ϕ : G/N → ‫ގ‬, by: ϕ(X) = |A ∩ X| where X = xN is a coset of N. Then: |(A ∩ X) (sA ∩ X)| + |(A ∩ X) (s−1 A ∩ X)| X∈G/H ≤ |A sA| + |A s−1 A| ≤ 2ε|A|. The L 1 -norm of ϕ is: X∈G/H ϕ(X) = |A|. So: |sϕ − ϕ|L 1 ≤ 2ε|ϕ|L 1 . Thus ϕ is an «almost invariant function» in the L 1 -norm. We have to show that there exists «an almost invariant» subset. For j ∈ {1, 2, 3, . } define B j = {X ∈ G/H | ϕ(X) ≥ j}.

The following theorem is due to Zimmer [Zi2]. We bring a somewhat different proof. Another proof can be found in [HV]. 7. If Γ is a finitely generated Kazhdan subgroup of SL 2 (‫)ރ‬, then Γ is finite. In particular, SU(2) (and hence also SO(n), n = 2, 3, 4) does not have a dense countable Kazhdan subgroup. The proof will be divided up to several propositions. Some of them are of independent interest. We are actually using in the proof some notions which are presented in more detail in Chapters 5 and 6.

Download PDF sample

Discrete Groups, Expanding Graphs and Invariant Measures by Alex Lubotzky, Jonathan D. Rogawski


by Kenneth
4.3

Rated 4.31 of 5 – based on 32 votes