Short talks

Thursday March 6
Timings	Name	Title	Abstract
16:00-16:10	Sumanta Das	Strong Topological Rigidity of Orientable Infinite-type Surfaces	We prove that if a homotopy equivalence between any two orientable infinite-type surfaces is a proper map, then it is properly homotopic to a homeomorphism. Thus, all orientable infinite-type surfaces are topologically rigid in a strong sense.
16:10-16:20	Apeksha Sanghi	Twisted Rokhlin property of mapping class groups	In this talk, we will see whether the big mapping class groups (MCG(S)) of surfaces (S) with empty boundary admits a dense $\phi$-twisted conjugacy class for all automorphisms $\phi \in Aut(MCG(S))$. This property is called the twisted Rokhlin property. This supplements the recent work of Lanier and Vlamis on the Rokhlin property of big mapping class groups. In the end, we will see the $R_{\infty}$ property of MCG(S).
16:25-16:35	Uddhab Roy	Theory of Fuchsian and semi-Fuchsian Schottky groups	In this talk, our goal is to initiate the study of estimates of the classical and non-classical Schottky structures in the discrete subgroups of the projective special linear group over the real numbers degree 2. In particular, after initiating the notion of classical Fuchsian Schottky group (different from the idea of real Schottky groups!), we investigate the non-classical generating sets in the Fuchsian Schottky groups in the upper-half plane with the circle at infinity as the boundary. After that, we introduce the notion of semi-Fuchsian Schottky groups in classical flavor. Finally, we derive a non-trivial example of the semi-Fuchsian Schottky group with non-classical generating sets within the hyperbolic plane.
16:35-16:45	Arkajit Pal Choudhury	Maximal Gromov hyperbolic spaces	The real hyperbolic spaces have a canonical cross-ratios on their boundary, and any Moebius homeomorphism on the boundary extends to an isometry on the space. Similarly, CAT(-1) spaces are known to have a canonical cross-ratio on the boundary with respect to the visual metrics. Isometry between CAT(-1) spaces extends to a Moebius homeomorphism (cross-ratio preserving homeomorphism) between their boundaries. An important open problem is the converse: Given a Moebius homeomorphism between the boundaries can we extend it to an isometry of CAT(-1) spaces?(This is the Mobius rigidity problem). The main motivation for this problem is the marked length spectrum rigidity problem . We describe some partial answers to the Moebius rigidity question. Biswas demonstrated that we have a positive answer to the Moebius problem for a special type of Gromov hyperbolic spaces called "maximal Gromov hyperbolic spaces". These maximal Gromov hyperbolic spaces satisfy a universal property: Any good (proper geodesically complete “boundary continuous”) Gromov hyperbolic space with the same boundary embeds isometrically into the maximal Gromov hyperbolic space. The boundary of a good Gromov hyperbolic space is a special type of compact space called "quasi-metric antipodal space". He also proved an equivalence of categories between maximal Gromov hyperbolic spaces and quasi-metric antipodal spaces. We will describe a structure theorem for maximal Gromov hyperbolic spaces with finite boundary. This is a joint work with Kingshook Biswas.
16:50-17:00	Rakesh Halder (online)	Landing rays and Cannon-Thurston maps	For a hyperbolic subgroup H of a hyperbolic group G, we describe sufficient criteria to guarantee the following two conditions a) Geodesic rays in H starting at 1 land at a unique point of the boundary of G b) The inclusion of H in G does not extend continuously to the boundary As a consequence we obtain diverse classes of examples demonstrating the non-existence of Cannon-Thurston maps. This is joint work with Mahan Mj and Pranab Sardar.
Friday March 7
11:30-11:40	Bhola Nath Saha	Length of filling pairs on punctured surfaces	A pair (α, β) of simple closed curves on an oriented surface $S_{g,n}$ of genus g and with n punctures is called a filling pair if the complement of the union of the curves is a disjoint union of topological disks with at most one puncture. In this talk, we study the lengths of filling pairs on once-punctured hyperbolic surfaces. In particular, we find a lower bound of the lengths of filling pairs. Furthermore, we show that this lower bound is the best one which depends only on the topology of the surface. This is a joint work with Dr. Bidyut Sanki.
11:40-11:50	Tathagata Nayak	On Character Variety of Anosov Representations	Let Γ be the fundamental group of an k-punctured, k ≥ 0, closed connected orientable surface of genus g ≥ 2. In this talk, it will be shown that the character variety of the (Q+, Q-)-Anosov irreducible representations, resp. the character variety of the (P +, P −)-Anosov Zariski dense representations of Γ into SL(n, C), n ≥ 2, is a complex manifold of complex dimension (2g + k − 2)(n^2 − 1). For Γ = π1(Σg ), these character varieties are holomorphic symplectic manifolds. This talk is based on a joint work with Prof. Krishnendu Gongopadhyay.
11:55-12:05	Debattam Das	Reciprocal elements in Picard group	An element $g$ in a group $G$ is called \emph{reciprocal} if there exists $h \in G$ such that $g^{-1}=hgh^{-1}$. The reciprocal elements are also known as `real elements' or `reversible elements' in the literature. In this talk, we will discuss about the reciprocal elements in Picard group. This is joint ongoing work with Krishnendu Gongopadhyay.
12:05-12:15	Sayantika Mondal	Distinguishing filling curve types via special metrics	In this talk, we look at filling curves on hyperbolic surfaces and consider its length infima in the moduli space of the surface as a type invariant. In particular explore the relations between the length infimum of curves and their self-intersection number. For any given surface, we will construct infinite families of filling curves that cannot be distinguished by self-intersection number but via length infimum. I might also discuss some coarse bounds on the metrics associated to these minimum lengths.
12:20 - 12:30	Deblina Das	Simple lift of non simple closed curves	Let $S$ is a connected, compact, oriented surface of finite genus $g$ and finitely many boundary components. It is a celebrated theorem of Peter Scott that each non-simple closed curve $\gamma$ lifts to a simple closed curve on some finite-sheeted cover. In this talk, we provide examples of finite-sheeted covers $\tilde{S}$ of $S$ and non-simple closed curves $\gamma$ on $S$ which lifts to simple closed curves on $\tilde{S}$. In particular, given any positive integer $n\geq 2$, we construct explicit non-simple closed curves on $S$ which has a simple lift to a degree $n$ cover of $S$. This gives a partial converse of Scott's theorem. This is a joint work with Dr. Arpan Kabiraj.
12:30-12:45	Pankaj Kapari	Generating the normalizer and the centralizer of spherical periodic mapping classes	For g ≥ 2, let Mod(S_g) be the mapping class group of closed, connected, and oriented surface S_g of genus g. In this talk, we will discuss a method to derive a finite generating set for the normalizer and the centralizer of a periodic mapping class F in Mod(S_g) having the orbit space genus 0. As an application, we derive a finite presentation for the centralizer and the normalizer of a reducible periodic mapping class in Mod(S_g) of the highest order 2g + 2. This is a joint work with Kashyap Rajeevsarathy and Apeksha Sanghi.
12:45-12:55	Pabitra Barman	Dominating surface-group representations in $PSL_n(\mathbb{C})$	Let $S$ be a punctured surface of negative Euler characteristic. In this talk, we show that a generic representation $\rho:\pi_1(S) \rightarrow PSL_n(\mathbb{C})$ is dominated by a positive representation $\rho_0:\pi_1(S) \rightarrow PSL_n(\mathbb{R})$ in the Hilbert length spectrum as well as in the translation length spectrum in the symmetric space $\mathbb{X}_n= \PSL_n(\mathbb{C})/\mathrm{PSU}(n)$, while preserving the lengths of peripheral curves. This result gives a new perspective on positivity and geometric structures in higher-rank Teichm\"{u}ller theory.