HU Credits:
2
Degree/Cycle:
2nd degree (Master)
Responsible Department:
Mathematics
Semester:
2nd Semester
Teaching Languages:
English and Hebrew
Campus:
E. Safra
Course/Module Coordinator:
Prof. Avinoam Mann
Coordinator Office Hours:
By appointment
Teaching Staff:
Prof Alex Lubotzky
Course/Module description:
This semester I’ll give a course titled ”Topics in group theory  growth”.
It will take place in room 209 on Tuesdays, 1113. It will follow broadly my book
”How groups grow”, but the details may be diﬀerent.
Given a group G generated by a ﬁnite set x(1); :::; x(d), we form all products
of n of the generators and their inverses, and let a(n) be the number of distinct
group elements obtained by this. a(n) is the growth function of G. It is at most
exponential, and if G is inﬁnite, at least linear. We are interested in properties of
this function, and its relation to the structure of G. Two seminal results in this
topic are
1. M.Gromov  the growth of G is (at most) polynomial iﬀ G contains a nilpotent
subgroup of ﬁnite index.
2. R.I.Grigorchuk  there exist groups whose growth is intermediate, i.e. faster
than any polynomial and slower than any exponential.
After several preliminary chapters we will concentrate on the proof of Gromov’s
theorem.
Course/Module aims:
same as in learning outcomes.
Learning outcomes  On successful completion of this module, students should be able to:
Ability to prove and apply the theorems presented in the course.
Ability to apply correctly the mathematical methodology in the context of the course.
Acquiring the fundamentals as well as basic familiarity with the field which will assist in the understanding of advanced subjects.
Ability to understanding and explain the subjects taught in the course.
Attendance requirements(%):
100
Teaching arrangement and method of instruction:
Lecture
Course/Module Content:
This semester I’ll give a course titled ”Topics in group theory  growth”.
It will take place in room 209 on Tuesdays, 1113. It will follow broadly my book
”How groups grow”, but the details may be diﬀerent.
Given a group G generated by a ﬁnite set x(1); :::; x(d), we form all products
of n of the generators and their inverses, and let a(n) be the number of distinct
group elements obtained by this. a(n) is the growth function of G. It is at most
exponential, and if G is inﬁnite, at least linear. We are interested in properties of
this function, and its relation to the structure of G. Two seminal results in this
topic are
1. M.Gromov  the growth of G is (at most) polynomial iﬀ G contains a nilpotent
subgroup of ﬁnite index.
2. R.I.Grigorchuk  there exist groups whose growth is intermediate, i.e. faster
than any polynomial and slower than any exponential.
After several preliminary chapters we will concentrate on the proof of Gromov’s
theorem.
Required Reading:
none
Additional Reading Material:
Lecturer's book
Course/Module evaluation:
End of year written/oral examination 50 %
Presentation 50 %
Participation in Tutorials 0 %
Project work 0 %
Assignments 0 %
Reports 0 %
Research project 0 %
Quizzes 0 %
Other 0 %
Additional information:
none
