Lev Glebsky
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 Aug 6 awarded Popular Question Jul 8 comment Fixed points of $x\mapsto 2^{2^{2^{2^x}}} \mod p$ Here are an estimates (From above and, I think, very imprecise)of number of solutions of $q^{x^{x^x}}=x \mod p$ based on similar constructions. There are some annoying details due to $x+k (\mod p)= x+k (\mod p-1)$ or $x+k(\mod p-1) +1$. So, $2^{x+k}= 2^x2^k$ or $22^x2^k\mod p$. I don't think that there is an easy generalization for points of period 4. I think that the proof goes through due to the group $\langle a,b,w\;|\;b^{-1}ab=b^2,w^{-1}aw=b, w^3=1\rangle$ is finite. (We may define "almost action" of this group, with $xa=x+1$, $xb=2x$, $xw=2^x\mod p$) Jul 8 comment Fixed points of $x\mapsto 2^{2^{2^{2^x}}} \mod p$ Here an estimates of Dec 14 comment A special residually finite group @IanAgol: did you mean some Wilson's construction of just infinte groups of type N(h)? Is it published now? Sep 26 revised The free group $F_2$ has index 12 in SL(2,$\mathbb{Z}$) edited body Sep 24 awarded Necromancer Sep 6 awarded Necromancer Mar 14 awarded Yearling Feb 3 answered Measures idempotent with respect to addition and multiplication. Feb 3 awarded Commentator Feb 3 comment The Higman group II @Ashot. This answers my first question! Thank you. Feb 3 awarded Scholar Feb 3 accepted The Higman group II Feb 2 revised The Higman group II added 45 characters in body Feb 2 comment The Higman group II @Yves and @Derek. Oops, you are right. I will make the corresponding corrections. Thank you. Feb 1 asked The Higman group II Jan 24 comment Eigenvalues of the products of a fixed unitari matrix with diagonal unitari matrices @ Michael I have not notice your comment before. I just consider it as a set... About b). Let $n$ be "very large" Then in example 2) al matrices $DU$ have "very small" spectral gaps. So, the question: for which $U$ all $DU$ have small maximal spectral gap? As $\\{DU\\}$ may be considered as a point in the flag manifold, one could try to relate this gap with a Reimann distance on the flag manifold.... Jan 10 revised Eigenvalues of the products of a fixed unitari matrix with diagonal unitari matrices added 89 characters in body Jan 9 asked Eigenvalues of the products of a fixed unitari matrix with diagonal unitari matrices Jan 9 comment A subgroup intersects every conjugacy class Yes, a free group seems to have a large subgroup. It may be constructed inductively adding $g$ to $$for each g with g^F\cap =\emptyset. (We need to start with a good initial$$ to avoid getting all $F$.)