## COLLOQUIA PATAVINA |

Avinoam Mann (Einstein Institute of Mathematics - Hebrew University of Jerusalem)

What is asymptotic group theory?

September 24, 2013

What is asymptotic group theory?

September 24, 2013

Abstract

Avinoam Mann is a Professor Emeritus at his Alma Mater, the Einstein In recent years there is much interest in various numerical questions, of asymptotic nature, in group theory.
Typical problems are:

1. How many groups there are of a given finite order?

2. Given a finitely generated group, how many elements can be obtained by multiplying the generators by each other not more than n times?

3. Given a finitely generated group, how many subgroups does it have of a given finite index (this number is finite)?

Some of these questions have geometrical motivations. The answers may require some of the deepest results in group theory such as the classification of the finite simple groups, and may lead to new methods, e.g. Gromov's theory of asymptotic cones.

1. How many groups there are of a given finite order?

2. Given a finitely generated group, how many elements can be obtained by multiplying the generators by each other not more than n times?

3. Given a finitely generated group, how many subgroups does it have of a given finite index (this number is finite)?

Some of these questions have geometrical motivations. The answers may require some of the deepest results in group theory such as the classification of the finite simple groups, and may lead to new methods, e.g. Gromov's theory of asymptotic cones.

Short bio

Avinoam Mann is a Professor Emeritus at his Alma Mater, the Einstein Institute of Mathematics in the Hebrew
University of Jerusalem. He has published over a hundred papers on group theory, giving very important contributions on several branches (p-groups, residually finite groups, growth and probabilistic aspects). He co-authored the influential book Analytic Pro-p Groups
and his recent book "How Groups Grow" is an excellent introduction to anyone interested in beginning a research project in this area.