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Pranab Sardar

Publications and source records attributed to Pranab Sardar.

16 recordsLinked to original sources
arXiv2026

A Characterization of Relative Hyperbolicity via Morse and Contracting Boundaries

We prove the following boundary-theoretic characterization of relatively hyperbolic groups. Let $G$ be a finitely generated group with a finite collection $\mathcal{H}$ of finitely generated subgroups, and let $G^h$ denote the associated cusped space. We prove that the pair $(G,\mathcal{H})$ is non-elementary relatively hyperbolic if and only if the Morse boundary $\partial_M^{\mathcal{DL}} G^h$ or the contracting boundary $\partial_c^{\mathcal{FQ}} G^h$ is non-empty and compact.

math.GT↗
arXiv2025

Embeddings of trees of hyperbolic metric spaces and Cannon--Thurston maps

Given a tree of hyperbolic metric spaces $\pi:X\to T$ a la Bestvina--Feighn (\cite{BF}), and a hyperbolic subspace $Y$ of $X$ with an induced tree of hyperbolic spaces structure over a subtree $S\subset T$, we address the question as to when the Cannon--Thurston (CT) map exists for the inclusion $Y\to X$. In this paper, we find additional sufficient conditions under which the CT map $\partial Y \to \partial X$ exists. However, we show with examples that this may fail to hold in general. These results about trees of spaces are then applied to graphs of hyperbolic groups to prove various existence results for CT maps. A very special instance of these results is the following: \emph{Suppose $G_1$ and $G_2$ are hyperbolic groups with a common quasiconvex subgroup $H$, and the free product with amalgamation $G = G_1 *_H G_2$ is hyperbolic. Suppose $K_i < G_i$, $i = 1,2$ are hyperbolic subgroups containing $H$ and $K=K_1*_H K_2$. Then $K$ (is hyperbolic and,) the inclusion $K\to G$ admits a CT map if the inclusions $K_i\to G_i$, $i=1,2$ admit CT maps.}

math.GT↗
arXiv2025

Landing rays and ray Cannon-Thurston maps

In this paper, we describe a procedure to construct pairs of hyperbolic groups $H<G$ with the following properties. 1) Every geodesic ray $\gamma$ in $H$ converges to a point $\xi_{\gamma}\in \partial G$. 2) The inclusion of $H$ into $G$ does not extend continuously to $\partial H \to \partial G$. In other words, a Cannon--Thurston map does not exist for this pair of hyperbolic groups. Jeon, Kapovich, Leininger and Ohshika gave a property of conical limit points in the presence of a Cannon--Thurston map. We convert this into a criterion for the existence of Cannon--Thurston maps and use it to prove the non-existence result in (2). We obtain, in particular, a geometric proof of Baker--Riley's counterexample.

math.GT↗
arXiv2023

On Geometry of Coned-Off Spaces and Cannon-Thurston Maps

A typical question addressed in this paper is the following. Suppose $Z\subset Y\subset X$ are hyperbolic spaces where $Z$ is quasiconvex in both $Y$ and $X$. Let $\HAT{Y}$ and $\HAT{X}$ denote the spaces obtained from $Y$ and $X$ respectively by coning off $Z$ as defined by Farb. {\em If the inclusion of the coned-off spaces $\HAT{Y}\map \HAT{X}$ admits the Cannon-Thurston (CT) map then does the inclusion $Y\map X$ also admit the Cannon-Thurston map?} The main result of this paper answers this question affirmatively provided $\HAT{Y}\map \HAT{X}$ satisfies Mitra's criterion for the existence of CT maps, although the answer in general is negative. The main application of our theorem is in the context of acylindrical complexes of hyperbolic groups. A. Martin proved a combination theorem for developable, acylindrical complexes of hyperbolic groups. Suppose $(\mathcal G, \YY)$ is an acylindrical complex of hyperbolic groups with universal cover $B$ which satisfy the hypotheses of Martin's theorem. Suppose $\YY_1\subset \YY$ is a connected subcomplex such that the subcomplex of groups $(\mathcal G, \YY_1)$ also satisfies the hypotheses of Martin's theorem, it has universal cover $B_1$ and the natural homomorphism $π_1(\mathcal G, \YY_1)\map π_1(\mathcal G, \YY)$ is injective. It follows from the main theorem of this paper that the inclusion $π_1(\mathcal G, \YY_1)\map π_1(\mathcal G, \YY)$ admits the CT map if the inclusion $B_1\rightarrow B$ satisfies Mitra's criterion. Also $π_1(\mathcal G, \YY_1)$ is quasiconvex in $π_1(\mathcal G, \YY)$ if in addition $B_1$ is qi embedded in $B$.

math.GR↗
arXiv2022

Trees of hyperbolic spaces

We give an alternative proof of the Bestvina--Feighn combination theorem for trees hyperbolic spaces and describe uniform quasigeodesics in such spaces. As one of the applications, we prove the existence of Cannon-Thurston maps for inclusion maps of total spaces of subtrees of hyperbolic spaces.

math.GR↗
arXiv2021

Research announcement: A combination theorem for acylindrical complexes of hyperbolic groups and Cannon-Thurston maps

This is an announcement of some of the results obtained as a part of the second author's Ph.D. thesis. In the first part, we prove that the fundamental group of an acylindrical complex of hyperbolic groups with finite edge groups is hyperbolic in which the vertex groups are quasiconvex. In the second part of the article, we prove the existence of Cannon-Thurston maps for certain subcomplexes of groups in acylindrical complexes of hyperbolic groups (see Theorem 0.4).

math.GR↗
arXiv2021

Propagating quasiconvexity from fibers

Let $1 \to K \longrightarrow G \stackrelπ\longrightarrow Q$ be an exact sequence of hyperbolic groups. Let $Q_1 < Q$ be a quasiconvex subgroup and let $G_1=π^{-1}(Q_1)$. Under relatively mild conditions (e.g. if $K$ is a closed surface group or a free group and $Q$ is convex cocompact), we show that infinite index quasiconvex subgroups of $G_1$ are quasiconvex in $G$. Related results are proven for metric bundles, developable complexes of groups, and graphs of groups.

math.GT↗
arXiv2020

Pullbacks of metric bundles and Cannon-Thurston maps

In this version of the paper the exposition is improved and gaps in some of the arguments filled following referee comments. We also include an appendix explaining the equivalence of flaring conditions for a metric bundle and the canonical metric graph bundle associated to it.

math.GT↗
arXiv2015

Existence and non-existence of bounded packing in CAT(0) spaces and Gromov hyperbolic spaces

The main result of this paper is that given a group $G$ acting geometrically by isometries on a CAT(0) space $X$ and a cyclic subgroup $H$ of $G$ generated by a rank-1 isometry of $X$, $H$ has bounded packing in $G$. We give two proofs of this result. The first one is by a characterization of rank-$1$ isometries by Hamenstadt. The second proof follows directly from some results of Dahmani-Guirardel-Osin and Sisto. Then using Mihailova's construction, we show the existence of a finitely generated subgroup of the direct product of two free groups $\mathbb F_2\times \mathbb F_2$ without the bounded packing property answering a question of Hruska-Wise. We also prove the existence of finitely presented subgroups of CAT(0) groups without bounded packing using Wise's {\em modified Rip's construction} and the {\bf 1-2-3} theorem of Baumslag, Bridson, Miller and Short.

math.GR↗
arXiv2014

Packing subgroups in solvable groups

We show that any subgroup of a (virtually) nilpotent-by-polycyclic group satisfies the bounded packing property of Hruska-Wise. In particular, the same is true about metabelian groups and linear solvable groups. However, we find an example of a finitely generated solvable group of derived length 3 which admits a finitely generated subgroup without the bounded packing property. In this example the subgroup is a metabelian retract also. Thus we obtain a negative answer to Problem 2.27 of Hruska-Wise. On the other hand, we show that polycyclic subgroups of solvable groups satisfy the bounded packing property.

math.GT↗
arXiv2012

A Combination Theorem for Metric Bundles

We define metric bundles/metric graph bundles which provide a purely topological/coarse-geometric generalization of the notion of trees of metric spaces a la Bestvina-Feighn in the special case that the inclusions of the edge spaces into the vertex spaces are uniform coarsely surjective quasi-isometries. We prove the existence of quasi-isometric sections in this generality. Then we prove a combination theorem for metric (graph) bundles (including exact sequences of groups) that establishes sufficient conditions, particularly flaring, under which the metric bundles are hyperbolic. We use this to give examples of surface bundles over hyperbolic disks, whose universal cover is Gromov-hyperbolic. We also show that in typical situations, flaring is also a necessary condition.

math.GT↗
arXiv2008

Projective normality of quotient varieties modulo finite groups

In this note, we prove that for any finite dimensional vector space $V$ over an algebraically closed field $k$, and for any finite subgroup $G$ of $GL(V)$ which is either solvable or is generated by pseudo reflections such that the $|G|$ is a unit in $k$, the projective variety $\mathbb P(V)/G$ is projectively normal with respect to the descent of $\mathcal O(1)^{\otimes |G|}$.

math.AG↗
arXiv2007

Torus quotients of homogeneous spaces of the general linear group and the standard representation of certain symmetric groups

We give a stratification of the GIT quotient of the Grassmannian $G_{2,n}$ modulo the normaliser of a maximal torus of $SL_{n}(k)$ with respect to the ample generator of the Picard group of $G_{2,n}$. We also prove that the flag variety $GL_{n}(k)/B_{n}$ can be obtained as a GIT quotient of $GL_{n+1}(k)/B_{n+1}$ modulo a maximal torus of $SL_{n+1}(k)$ for a suitable choice of an ample line bundle on $GL_{n+1}(k)/B_{n+1}$.

math.AG↗