March 2021
Talk 9
Date: 02 March 2021, 16:00 Indian Standard Time (10:30AM GMT)
Speaker: Dr. Pradeep Rai, HRI Prayagraj, India
Title: Schur multiplier of prime power groups.
Abstract: Schur multiplier of a finite group G is defined as the second cohomology group of G with coefficients in the multiplicative group of complex numbers considering the trivial action of G. This originated in a work of Issai Schur on projective representations of groups and plays an important role in the theory of extensions. We shall give a survey talk on the Schur multiplier of prime power groups with a focus on the bounds for the order the Schur multiplier.
Talk 10
Date: 09 March 2021, 16:00 Indian Standard Time (10:30AM GMT)
Speaker: Dr. Abhijit Pal, IIT Kanpur, India
Title: Height of subgroups.
Abstract: Hyperbolic groups and relatively hyperbolic groups, introduced by M. Gromov, captures the geometry of a group acting on a non-positively curved space. Height of a subgroup in these groups roughly measures the total intersection of conjugates of the subgroup. In this talk, after defining related basic concepts, I will focus on the recent developments made on the height of subgroups.
Talk 11
Date: 16 March 2021, 16:00 Indian Standard Time (10:30AM GMT)
Speaker: Dr. Kashyap Rajeevsarathy, IISER Bhopal, India
Title: Liftable mapping class groups of regular cyclic covers.
Abstract: Let Mod(S_g) be the mapping class group of the closed orientable surface of genus g \geq 1. In this talk, we will consider the standard k-sheeted regular cover p_k : S_{k(g-1)+1} to S_g for k \geq 2, and analyze the subgroup LMod_{p_k}(S_g) of mapping classes that lift under the cover p_k. We will show that LMod_{p_k}(S_g) is the stabilizer subgroup of Mod(S_g) with respect to a collection of vectors in H_1(S_g,\mathbb{Z}_k), and also derive a symplectic criterion for the liftability of a given mapping class under p_k. As an application of this criterion, we will obtain a normal series of LMod_{p_k}(S_g), which generalizes a well known normal series of the congruence subgroup Gamma_0(k) of SL(2,\mathbb{Z}). Among other applications, we will describe a procedure for obtaining a finite generating set for LMod_{p_k}(S_g).
PDF of Abstract Notes of the talk Video of the talk
Talk 12
Date: 23 March 2021, 16:00 Indian Standard Time (10:30AM GMT)
Speaker: Dr. Krishnendu Gongopadhyay, IISER Mohali, India
Title: Real Unipotent Elements in Simple Lie Groups.
Abstract: Real elements are those elements in a group which are conjugate to their own inverses. Real elements appear naturally at different branches of mathematics. These elements are also known as `reversible' elements in the literature. These elements are closely related to the so-called strongly real elements in a group which are products of two involutions. After giving a brief exposition on real elements in groups, I shall discuss classification of real unipotent elements in simple Lie groups which is part of a joint work with Chandan Maity.
Notes of the talk Video of the talk
Talk 13
Date: 30 March 2021, 16:00 Indian Standard Time (10:30AM GMT)
Speaker: Professor Mahan Mj, TIFR Mumbai, India
Title: Percolation on Hyperbolic groups
Abstract: We study first passage percolation (FPP) in a Gromov-hyperbolic group G with boundary equipped with the Patterson-Sullivan measure. We associate an i.i.d. collection of random passage times to each edge of a Cayley graph of G, and investigate classical questions about asymptotics of first passage time as well as the geometry of geodesics in the FPP metric. Under suitable conditions on the passage time distribution, we show that the 'velocity' exists in almost every direction, and is almost surely constant by ergodicity of the G-action on the boundary.
For every point on the boundary, we also show almost sure coalescence of any two geodesic rays directed towards the point. Finally, we show that the variance of the first passage time grows linearly with word distance along word geodesic rays in every fixed boundary direction.
This is joint work with Riddhipratim Basu.