My Research

## Research Interests

Low dimensional Topology :

Let G be a finite group. A spectrum sp(G) of G is a collection of
non-negative integers g such that there is a Riemann Surface X of
genus g on which G acts via orientation preserving diffeomorphisms. My aim is to give a description of the contravariant
functor G → sp(G). It is also known as Hurwitz Problem.

Ramanujan Graphs :

Cayley graphs of finite groups well studied since last 50 years. The Ramanujan behaviour of these graphs determines how well
connected these graphs are. This question connects to the arithmetics of finite algebraic objects, modular forms and local
measure theory of finite graphs. For example, in case of metacyclic groups, the underlying base rings play a crucial role.
I am interested to know the Ramanujan properties for various finite groups.
# Publications

On the genus spectrum for p-groups
of exponent p and p-groups of maximal class, J. Group Theory, Vol 12, No. 1, 2009, Pg.39-54.

A structured description of the genus spectrum of abelian p-groups
(with Juergen Mueller) Glasgow Math. J., Vol 61, No. 2, 2019, Pg.381-423.

[Journal version]
Spectral properties of the Cayley Graphs of split metacyclic groups (with K. Rajeevsarathy, P. Aurora, S. Lakshmivarahan) J. Ramanujan Math. Soc., Vol. 35, No. 2, 2020, Pg.159-175.

[Journal version]
Bound on the diameter of split metacyclic groups (with Kashyap Rajeevsarathy) J. Algebra Appl., Vol. 19, No. 11, 2020.

[Journal version]
A survey on the Non-inner Automorphism Conjecture (with Renu) Math. Newsletter, J. Ramanujan Math. Soc., Vol. 31, No. 1, 2020, Pg.18-24.

[Journal version]
Lambda Numbers of Finite p-Groups (with Mayank Mishra) J. Algebraic Combin., Vol. 57, No. 1, 2023. Pg.101-110.
[Journal version]
Finite symmetries of surfaces of p-groups of co-class 1, J. Algebra Appl., Vol. 22, No. 6, 2023 Pg. 2350122-(1-33)
[Journal version]
On the Schur Multiplier of finite p-groups of maximal class (with Renu) J. Group Theory, Vol. 26, No. 3, 2023, Pg.533-545.
[Journal version]
A Note on a Conjecture of Gao and Zhuang for groups of order 27 (with Naveen Kumar) (accepted for publication in J. Algebra Appl.).
# Preprints

# PhD students

- Renu Joshi : Finite p-groups.
- Naveen Kumar : Algebraic number theory.

# Mathscinet Review