here - Royal Holloway

RESEARCH STATEMENT
THEOFANIS ALEXOUDAS
1. My Thesis
My thesis [1] is concerned with the study of maximal subgroups of torsion multi-edge spinal groups.
The main result states that every torsion multi-edge spinal group has maximal subgroups only of
finite index. This implies that every such group does not contain dense proper subgroups with
respect to the profinite topology. Moreover, for a torsion multi-edge spinal group, I showed that all
its maximal subgroups are normal of finite index p, where p is the odd prime such that the group
acts on the p-adic regular rooted tree.
Multi-edge spinal groups are modelled after the GGS-groups, named after R. Grigorchuk, N. Gupta
and S. Sidki. Every group in the class of GGS-groups is generated by a rooted automorphism
and a recursively defined directed automorphism. In contrast to the GGS-groups, every multi-edge
spinal group is generated by a rooted automorphism and an arbitrary finite number of directed
automorphisms.
2. Short-Term Research Goals
2.1. Commensurability of subgroups. In [6], R. Grigorchuk and J. Wilson established some
interesting results concerning abstract commensurability of subgroups for the first Grigorchuk group.
More precisely, the authors showed that if a group L is commensurable with the first Grigorchuk
group, then all maximal subgroups of L have finite index in L. Their result applies to GGS-groups,
and it would be very interesting to investigate this property for the class of spinal groups.
2.2. Maximal subgroups. Pervova [10, 12] showed that the Grigorchuk group and torsion GGSgroups do not contain maximal subgroups of infinite index. Equivalently, these groups do not contain
proper dense subgroups with respect to the profinite topology. On the other hand, prompted by a
´ [3], Bondarenko gave in [4] a non-constructive example
ˇ k
question of Grigorchuk, Bartholdi and Suni
of a finitely generated branch group that does have maximal subgroups of infinite index. Hence we
face the following problem.
Problem 2.1. Characterize among finitely generated branch groups those that possess maximal
subgroups of infinite index and those that do not.
2.3. The Congruence Subgroup Property. A multi-edge spinal group G is said to possess the
congruence subgroup property if there exists some n ∈ N such that every subgroup of finite index
in G contains a level stabilizer. In other words, if the profinite topology and the congruence topology
coincide.
Pervova [9] showed that the Grigorchuk group and GGS-groups do possess the congruence subgroup
property. On the other hand, the author [11] constructed torsion multi-edge spinal groups which do
not possess the congruence subgroup property. Hence we face the following problem.
Problem 2.2. Characterize among torsion multi-edge spinal groups those that possess the congruence subgroup property and those that do not.
date: March 31, 2014.
2010 Mathematics Subject Classification. Primary 20E08, 20E18, 20E28; Secondary 20E07, 20E45, 20F50, 20F65.
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THEOFANIS ALEXOUDAS
3. Long-Term Research Goals
3.1. Representation growth and representation zeta functions of groups. The study of
representation growth and representation zeta functions of groups (or other asymptotic invariants
of finitely generated infinite groups) has a long and celebrated history; see [7, 14] for two excellent
introductions to the subject.
In the context of multi-edge spinal groups, it is only recently that Bartholdi [2] initiated a study of
representation zeta functions of self-similar branched groups. It would be very desirable to develop
a representation theory for such groups and study representation zeta functions associated to multiedge spinal groups.
3.2. Graphs associated with conjugacy classes and character degrees. The study of conjugacy classes and character degrees is a well-established research area in finite group theory; see [5]
for a nice survey. In recent decades various mathematicians initiated a study of graphs associated
with character degrees and conjugacy classes. This line of investigation has often revealed important results about the structure of finite groups. Moreover, surprising connections have been found
between the type of graphs one can associate with invariants such as the character degrees and
conjugacy classes of finite groups, and structural properties of the groups in question; see [8] for
an overview of the subject. It would be very interesting to initiate a systematic study of graphs
associated to character degrees and conjugacy classes for finitely generated (abstract) just infinite
groups.
4. Conclusion
Although my mathematical home base is Geometric Group Theory to some extent, I am always
happy to learn about new things. In particular, my broader research interests include problems
in p-adic groups, Lie theory, expander graphs and others. I hope to be given the opportunity to
continue working on my long-standing and everlasting love of my life, that is Mathematics.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
T. Alexoudas, Maximal subgroups of branch groups, PhD thesis, Royal Holloway, University of London, 2013.
L. Bartholdi, Representation zeta functions of self-similar branch groups, arXiv:1303.1805 [math.GR].
´ Handbook of algebra 3, North-Holland, Amsterdam, 2003.
ˇ
L. Bartholdi, R. I. Grigorchuk and Z. Suni
k,
I. V. Bondarenko, Finite generation of iterated wreath products, Arch. Math. (Basel) 95(4) (2010), 301–308.
A. R. Camina and R. D. Camina, The influence of conjugacy class sizes on the structure of finite groups: a
survey, arXiv:1002.3960 [math.GR].
R. I. Grigorchuk and J. S. Wilson, A structural property concerning abstract commensurability of subgroups, J.
London Math. Soc. 68(2) (2003), 671–682.
B. Klopsch, Representation growth and representation functions of groups, arXiv:1209.2896 [math.GR].
M. L. Lewis, An overview of graphs associated with character degrees and conjugacy class sizes in finite groups,
Rocky Mountain Journal of Mathematics 38 (1) (2008), 175–211.
E. L. Pervova, Profinite topologies in just infinite branch groups, preprint of the Max Planck Institute for
Mathematics 154, Bonn, Germany, 2000.
E. L. Pervova, Everywhere dense subgroups of a group of tree automorphisms, Tr. Mat. Inst. Steklova 231 (Din.
Sist., Avtom. i. Beskon. Gruppy) (2000), 356–367.
E. L. Pervova, The congruence property of AT-groups, Algebra Logika 41 (5) (2002), 553–567.
E. L. Pervova, Maximal subgroups of some non locally finite p-groups, Internat. J. Algebra Comput. 15(5-6)
(2005), 1129–1150.
E. L. Pervova, Profinite completions of some groups acting on trees, J. Algebra 310 (2) (2007), 858–879.
C. Voll, A newcomer’s guide to zeta functions of groups and rings, arXiv:0906.1832 [math.GR].
Department of Mathematics, Royal Holloway University of London, Egham, TW20 0EX, United Kingdom
E-mail address: [email protected]

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