False proof of amenability and non-amenability of the R. Thompson group appear about once a year. The interesting thing is that about half of the wrong papers claim amenability and about half claim non-amenability. The latest attempt to prove non-amenability was made by Bronislaw Wajnryb and Pawel Witowicz. The idea of their argument is very nice. […]

This is a yet another attempt to do a blog post on the proof of Hilbert Nullstellensatz (another attempt is Terry Tao’s blog). Since I am lecturing in a few days on the Nullstellensatz, I decided to find the easiest proof, and discovered that there are at least 15 more or less different proofs of […]

Together with Gili Golan we wrote another paper on . Note that  Savchuk proved that for every number in the subgroup of all elements of that fix is maximal (and its Schreier graph is amenable). He asked if has any other maximal subgroups of infinite index. That problem is soved (in the negative) in our paper. In fact […]

Let be a countable group with a length function . That means . For example, a finitely generated group and its word length. Property Rapid Decay (introduced by Haagerup and Jolissaint) says that the operator norm of an element of the group algebra is not much bigger than its -norm. The property is interesting because […]

A proof from my book. This theorem was needed to estimate (from below) the growth function of Okninski’s semigroup. For every natural number let   denote the number of primes . Say, , , , etc. The next theorem was proved by Chebyshev in 1850. We present a proof based on some ideas of Erdos (he […]

The  paper “The Tarski numbers of groups” by Mikhail Ershov from Virginia, Gili Golan from Bar Ilan and myself can be found here. Here is an abstract: The Tarski number of a non-amenable group is the minimal number of pieces in a paradoxical decomposition of . In this paper we investigate how Tarski numbers may […]

Here is a new paper of mine. The story is this. A couple of years ago I was reading the paper  (the first version). I was surprised finding there a statement answering a 30-years old question of van den Dries and Wilkie (if one asymptotic cone of a group is locally compact, should the group be […]

I added a section about amenability (5.8), corrected many misprints. Here is the latest version of the book.

Most probably I won’t be active on mathoverflow.net any longer: too many homework level  questions and the community attitude towards these questions has become too liberal.  I think I did help a few people with my answers, and I got some useful information from the answers to my questions. 

Here is an adaptation of Lang’s proof of Gelfond’s theorem to prove that is not rational – needed for http://mathoverflow.net/questions/138247/prove-that-sqrt2-sqrt2-is-an-irrational-number-without-using-a-theorem/138291#138291. Consider the functions . Let , where is big enough (we shall see how large should be below). Suppose that is rational. Then is a rational number for every . Lemma 1. Let us have […]