arXiv subjects
Publications and source records attributed to Mahan Mj.
We prove that any one-relator group $G$ is the fundamental group of a compact Sasakian manifold if and only if $G$ is either finite cyclic or isomorphic to the fundamental group of a compact Riemann surface of genus g > 0 with at most one orbifold point of order $n \geq 1$. We also classify all groups of deficiency at least two that are also the fundamental group of some compact Sasakian manifold.
Answering a question due to Min, we prove that a finite graph of roses admits a regluing such that the resulting graph of roses has hyperbolic fundamental group.
We characterise completely when limit sets, as parametrised by Cannon-Thurston maps, move discontinuously for a sequence of algebraically convergent quasi-Fuchsian groups.
We study horospheres in hyperbolic 3-manifolds $M$ all whose ends are degenerate. Towards this, we study which almost minimizing geodesics in $M$ go through arbitrarily thin parts.
We generalize the notion of tight geodesics in the curve complex to tight trees. We then use tight trees to construct model geometries for certain surface bundles over graphs. This extends some aspects of the combinatorial model for doubly degenerate hyperbolic 3-manifolds developed by Brock, Canary, and Minsky during the course of their proof of the Ending Lamination Theorem. Thus we obtain uniformly Gromov-hyperbolic geometric model spaces equipped with geometric $G-$actions, where $G$ admits an exact sequence of the form $$1 \to π_1(S) \to G \to Q \to 1.$$ Here $S$ is a closed surface of genus $g > 1$ and $Q$ belongs to a special class of free convex cocompact subgroups of the mapping class group $MCG(S)$.
We determine the image of the monodromy map for meromorphic projective structures with poles of orders greater than two. This proves the analogue of a theorem of Gallo-Kapovich-Marden, and answers a question of Allegretti and Bridgeland. Our proof uses coordinates on the moduli space of framed representations arising from the work of Fock and Goncharov.
We introduce notions of Cheeger constants for graphons and graphings. We prove Cheeger and Buser inequalities for these. On the way we prove co-area formulae for graphons and graphings.
In this partly expository monograph we develop a general framework for producing uncountable families of exotic actions of certain classically studied groups acting on the circle. We show that if $L$ is a nontrivial limit group then the nonlinear representation variety $\mathrm{Hom}(L,\mathrm{Homeo}_+(S^1))$ contains uncountably many semi-conjugacy classes of faithful actions on $S^1$ with pairwise disjoint rotation spectra (except for $0$) such that each representation lifts to $\mathbb{R}$. For the case of most Fuchsian groups $L$, we prove further that this flexibility phenomenon occurs even locally, thus complementing a result of K. Mann. We prove that each non-elementary free or surface group admits an action on $S^1$ that is never semi-conjugate to any action that factors through a finite--dimensional connected Lie subgroup in $\mathrm{Homeo}_+(S^1)$. It is exhibited that the mapping class groups of bounded surfaces have non-semi-conjugate faithful actions on $S^1$. In the process of establishing these results, we prove general combination theorems for indiscrete subgroups of $\mathrm{PSL}_2(\mathbb{R})$ which apply to most Fuchsian groups and to all limit groups. We also show a Topological Baumslag Lemma, and general combination theorems for representations into Baire topological groups. The abundance of $\mathbb{Z}$--valued subadditive defect--one quasimorphisms on these groups would follow as a corollary. We also give a mostly self-contained reconciliation of the various notions of semi-conjugacy in the extant literature by showing that they are all equivalent.
Let $M$ be a compact complex manifold of dimension at least three and $Π: M\rightarrow X$ a positive principal elliptic fibration, where $X$ is a compact Kähler orbifold. Fix a preferred Hermitian metric on $M$. In \cite{V}, the third author proved that every stable vector bundle on $M$ is of the form $L\otimes Π^*B_0$, where $B_0$ is a stable vector bundle on $X$, and $L$ is a holomorphic line bundle on $M$. Here we prove that every stable Higgs bundle on $M$ is of the form $(L\otimes Π^*B_0,Π^*Φ_X)$, where $(B_0, Φ_X)$ is a stable Higgs bundle on $X$ and $L$ is a holomorphic line bundle on $M$.
We introduce the notions of geometric height and graded (geometric) relative hyperbolicity in this paper. We use these to characterize quasiconvexity in hyperbolic groups, relative quasiconvexity in relatively hyperbolic groups, and convex cocompactness in mapping class groups and $Out(F_n)$. Corrigendum: there is an unfortunate mistake in the statement and the proof of Proposition 5.1. This affects one direction of the implications of the main theorem. A correction is given, that states that given a quasi-convex subgroup of a hyperbolic (or relatively hyperbolic) group, the graded relative hyperbolic structure holds with respect to saturations of I-fold intersections, that are stabilizers of limit sets of I-fold intersections.
We explicate a number of notions of algebraic laminations existing in the literature, particularly in the context of an exact sequence $$1\to H\to G \to Q \to 1 $$ of hyperbolic groups. These laminations arise in different contexts: existence of Cannon-Thurston maps; closed geodesics exiting ends of manifolds; dual to actions on $\R-$trees. We use the relationship between these laminations to prove quasiconvexity results for finitely generated infinite index subgroups of $H$, the normal subgroup in the exact sequence above.
Let $M$ be a quasi-regular compact connected Sasakian manifold, and let $N = M/S^1$ be the base projective variety. We establish an equivalence between the class of Sasakian $G$-Higgs bundles over $M$ and the class of parabolic (or equivalently, ramified) $G$-Higgs bundles over the base $N$.
We give an overview of the theory of Cannon-Thurston maps which forms one of the links between the complex analytic and hyperbolic geometric study of Kleinian groups. We also briefly sketch connections to hyperbolic subgroups of hyperbolic groups and end with some open questions.
We show that Cannon-Thurston maps exist for degenerate free groups without parabolics, i.e. for handlebody groups. Combining these techniques with earlier work proving the existence of Cannon-Thurston maps for surface groups, we show that Cannon-Thurston maps exist for arbitrary finitely generated Kleinian groups without parabolics, proving conjectures of Thurston and McMullen. We also show that point pre-images under Cannon-Thurston maps for degenerate free groups without parabolics correspond to end-points of leaves of an ending lamination in the Masur domain, whenever a point has more than one pre-image. This proves a conjecture of Otal. We also prove a similar result for point pre-images under Cannon-Thurston maps for arbitrary finitely generated Kleinian groups without parabolics.
Let $(S,\, \ast)$ be a closed oriented surface with a marked point, let $G$ be a fixed group, and let $ρ\colonπ_1(S) \longrightarrow G$ be a representation such that the orbit of $ρ$ under the action of the mapping class group $Mod(S,\, \ast)$ is finite. We prove that the image of $ρ$ is finite. A similar result holds if $π_1(S)$ is replaced by the free group $F_n$ on $n\geq 2$ generators and where $Mod(S,\, \ast)$ is replaced by $Aut(F_n)$. We thus resolve a well-known question of M. Kisin. We show that if $G$ is a linear algebraic group and if the representation variety of $π_1(S)$ is replaced by the character variety, then there are infinite image representations which are fixed by the whole mapping class group.
We show that a finite collection of stable subgroups of a finitely generated group has finite height, finite width and bounded packing. We then use knowledge about intersections of conjugates to characterize finite families of quasimorphisms on hyperbolically embedded subgroups that can be to simultaneously extended to the ambient group.
We extend the Donaldson-Corlette-Hitchin-Simpson correspondence between Higgs bundles and flat connections on compact Kähler manifolds to compact quasi-regular Sasakian manifolds. A particular consequence is the translation of restrictions on Kähler groups proved using the Donaldson-Corlette-Hitchin-Simpson correspondence to fundamental groups of compact Sasakian manifolds.
Given an almost complex structure on a subbundle of the cotangent bundle, we prove a Castelnuovo--de Franchis type theorem for it.