**Institution**: Universidad Autónoma de Madrid

**Position**: Profesor Titular de Universidad

**Office**: 202 **Phone**:+34 912999 742

**E-mail**: andrei.jaikin()uam.es

#### Biographical Review

Andrei Jaikin-Zapirain is an associate professor of Department of Mathematica of the Autonomous University of Madrid and a memver of the Institute of Mathematics. He was born in Moscow in 1972. I 1995 he finished his undergraduate studies in the Faculty of Mechanics and Mathematics of the Moscow State University. He achieved his Ph.D. in 2001 at the University of the Basque Country.

The main interests of Andrei Jaikin-Zapirain belong to the field of Group Theory and concretely to the theory of finite and profinite groups. He made significant contributions to finite \(p\)-groups, finite groups with authomorphisms with few fixed points, the representation and subgroup growth, the verbal width of pro-\(p\) groups, groups with Kazhdan's property and Golod-Shafarevich groups. The most imprortant of his achievements are the following:

In (J. Amer. Math. Soc. 19 (2006), no. 1, 91--118) Andrei Jaikin-Zapirain proved that the representation zeta function \(\zeta^G(s)=\displaystyle \sum_{\lambda\in Irr(G)}\lambda(1)^{-s}\) of a compact FAb \(p\)-adic group \(G\), when \(p\)is odd, is equal to a rational function of \(\{n^{-s}:\ n\in\mathbb N\}\).

In a joint paper with Laci Pyber (Ann. of Math. (2) 173 (2011), no. 2, 769--814) a new method to counting finite permutation groups was introduced. Their main result shows that the number of conjugacy classes of \(d\)-generated primitive subgroups of \(\Sigma_n\) is at most \(n^{cd}\) for some absolute constant \(c\). As a consequence several open problems concerning random generation and the subgroup growth of profinite groups were solved.

In a joint paper with Mikhail Ershov (Invent. Math. 179 (2010), no. 2, 303--347) a new combinatorial criterion for Kazhdan's property (T) were discovered. This criterion is applicable to a large class of discrete groups. This method allowed to solve an old open problem: the group \(EL_n(R)\) with coefficients in a finitely generated unitary ring \(R\) has Kazhdan's property (T) if \(n>2\).

In (Adv. Math. 227 (2011), no. 3, 1129--1143) Andrei Jaikin-Zapirain established a super-logarithmic lower bound for the number of conjugacy classes of a finite nilpotent group.