arXiv subjects

Abhijit Pal

Publications and source records attributed to Abhijit Pal.

11 recordsLinked to original sources

arXiv2026

On 2-Sphere Bowditch Boundaries Attaining Conformal Dimension

Bonk and Kleiner proved that if $G$ is a Gromov hyperbolic group whose boundary $\partial_{\infty}G$ is homeomorphic to an Ahlfors $Q$-regular metric $2$-sphere $Z$, and the Ahlfors regular conformal dimension of $Z$ is attained and equal to $Q$, then $G$ acts discretely, cocompactly, and isometrically on $\mathbb{H}^3$. In this article, we extend the Bonk-Kleiner theorem to the setting of relatively hyperbolic groups. More precisely, we prove that if $(G,\mathcal{H})$ is a relatively hyperbolic group whose Bowditch boundary is homeomorphic to an Ahlfors $Q$-regular metric $2$-sphere $Z$, with the Ahlfors regular conformal dimension of $Z$ attained and equal to $Q$, then $G$ acts discretely and isometrically on $\mathbb{H}^3$, and every subgroup in $\mathcal{H}$ is virtually $\mathbb Z^2$.

math.GR↗

arXiv2024

Maps between Boundaries of Relatively Hyperbolic Groups

F. Paulin proved that if the Gromov boundaries of two hyperbolic groups are quasi-Mobius equivalent, then the groups themselves are quasi-isometric. The goal of this article is to extend Paulin's result to the setting of relatively hyperbolic groups by introducing the notion of relative quasi-Mobius maps between the Bowditch boundaries of relatively hyperbolic groups. We show that any coarsely cusp-preserving quasi-isometry between two relatively hyperbolic groups induces a homeomorphism between their Bowditch boundaries, and that this induced homeomorphism is relative quasi-Mobius and linearly distorts the exit points of bi-infinite geodesics into combinatorial horoballs. Conversely, we prove that if a homeomorphism between the Bowditch boundaries of two relatively hyperbolic groups preserves parabolic fixed points and is either relative quasi-Mobius or linearly distorts the exit points of bi-infinite geodesics into combinatorial horoballs, then it arises from a coarsely cusp-preserving quasi-isometry between the groups.

math.GT↗

arXiv2021

Strongly Contracting Geodesics in a Tree of Spaces

Let X be a tree of proper geodesic spaces with edge spaces strongly contracting and uniformly separated from each other by a number depending on the contraction function of edge spaces. Then we prove that the strongly contracting geodesics in vertex spaces are quasiconvex in X. We further prove that in X if all the vertex spaces are uniformly hyperbolic metric spaces then X is a hyperbolic metric space and vertex spaces are quasiconvex in X.

math.GR↗

arXiv2021

Contracting Boundary of a Cusped Space

Let $G$ be a finitely generated group. Cashen and Mackay proved that if the contracting boundary of $G$ with the topology of fellow travelling quasi-geodesics is compact then $G$ is a hyperbolic group. Let $\mathcal{H}$ be a finite collection of finitely generated infinite index subgroups of $G$. Let $G^h$ be the cusped space obtained by attaching combinatorial horoballs to each left cosets of elements of $\mathcal {H}$. In this article, we prove that if the combinatorial horoballs are contracting and $G^h$ has compact contracting boundary then $G$ is hyperbolic relative to $\mathcal{H}$.

math.GR↗

arXiv2020

Acylindrical Hyperbolicity of Subgroups

Suppose $G$ is a finitely generated group and $H$ is a subgroup of $G$. Let $\partial_{c}^{\mathcal{F}\mathcal{Q}}G$ denote the contracting boundary of $G$ with the topology of fellow travelling quasi-geodesics defined by Cashen-Mackay \cite{cashen2017}. In this article, we show that if the limit set $Λ(H)$ of $H$ in $\partial_{c}^{\mathcal{F}\mathcal{Q}}G$ is compact and contains at least three points then the action of the subgroup $H$ on the space of distinct triples $Θ_{3}(Λ(H))$ is properly discontinuous. By applying a result of B. Sun \cite{BinSun}, if the limit set $Λ(H)$ is compact and the action of $H$ on $\partial_{c}^{\mathcal{F}\mathcal{Q}}G$ is non-elementary then $H$ becomes an acylindrically hyperbolic group

math.GT↗

arXiv2020

Height in Splitting of Relatively Hyperbolic Groups

Given a finite graph of relatively hyperbolic groups with its fundamental group relatively hyperbolic and edge groups quasi-isometrically embedded and relatively quasiconvex in vertex groups, we prove that vertex groups are relatively quasiconvex if and only if all the vertex groups have finite relative height in the fundamental group.

math.GT↗

arXiv2018

Complex of Relatively Hyperbolic Groups

In this article, we prove a combination theorem for a complex of relatively hyperbolic groups. It is a generalization of Martin's \cite{martin} work for combination of hyperbolic groups over a finite $M_K$-simplicial complex, where $k\leq 0$.

math.GT↗

arXiv2011

Notes on Stable Teichmuller quasigeodesics

In this note, we prove that for a cobounded,Lipschitz path $γ:I\to\TT$, if the pull back bundle $\mathcal H_γ$ over $I$ is a strongly relatively hyperbolic metric space then there exists a geodesic $ξ$ in $\TT$ such that $γ(I)$ and $ξ$ are close to each other.

math.GT↗

arXiv2010

Relative Hyperbolicity, Trees of Spaces and Cannon-Thurston Maps

We prove the existence of continuous boundary extensions (Cannon-Thurston maps) for the inclusion of a vertex space into a tree of (strongly) relatively hyperbolic spaces satisfying the qi-embedded condition. This implies the same result for inclusion of vertex (or edge) subgroups in finite graphs of (strongly) relatively hyperbolic groups. This generalises a result of Bowditch for punctured surfaces in 3 manifolds and a result of Mitra for trees of hyperbolic metric spaces.

math.GR↗

arXiv2008

Relative Hyperbolic Extensions of Groups and Cannon-Thurston Maps

Let $1\to (K,K_1)\to (G,N_G(K_1))\to(Q,Q_1)\to 1$ be a short exact sequence of pairs of finitely generated groups with $K$ strongly hyperbolic relative to proper subgroup $K_1$. Assuming that for all $g\in G$ there exists $k\in K$ such that $gK_1g^{-1}=kK_1k^{-1}$, we prove that there exists a quasi-isometric section $s\colon Q \to G$. Further we prove that if $G$ is strongly hyperbolic relative to the normalizer subgroup $N_G(K_1)$ and weakly hyperbolic relative to $K_1$, then there exists a Cannon-Thurston map for the inclusion $i\colonΓ_K\to Γ_G$.

math.GR↗

arXiv1997

Study Of A Planar Lattice Model With $P_4$ Interaction

A planar square lattice model with 3-d spins interacting with nearest neighbours through a potential -$εP_4 (cos θ_{ij})$ is studied by Monte Carlo technique. Lattice sizes from 10$\times$10 to 30$\times$30 are considered for calculating various thermodynamic averages. A 80$\times$80 lattice has been used to obtain the pair correlation function. To accurately ascertain the order of the phase transition the Ferrenberg-Swendsen technique has been used on a 120$\times$120 lattice. Our study predicts that the system exhibits a first order phase transition which is confirmed by the twin-peaked nature of the distribution function. The pair correlation function shows an algebraic decay at low temperatures and an exponential decay at high temperatures. Mean field and Two-site Cluster calculations have also been performed and the latter is found to predict the thermodynamic averages fairly accurately.

cond-mat.soft↗