arXiv subjects
Publications and source records attributed to Mahan Mj.
We prove the existence of strongly converging unitary representations in various new settings of countable groups, in particular, arising as extensions by exact groups: semidirect products with amenable groups; generalized wreath products with abelian base; graph wreath products; various free-by-cyclic groups including Gersten's group; and general Bernoulli shift crossed products on $C^*$-algebras.
We start by surveying known properties of first passage percolation (FPP) geodesics on a Gromov-hyperbolic group $G$. Due to a coalescence phenomenon established in earlier work of the authors, a random tree $T(\xi,\omega)$ consisting of infinite random geodesics to a point $\xi$ on the Gromov boundary $\partial G$ emerges naturally. These trees have a rich random geometry that we explore further in this paper.
In this note, we construct algebraic correspondences as matings of Fuchsian $(p,q,\infty)$-triangle groups with Blaschke products. Combined with the results of [MM25], this proves mateability of all cusped triangle groups with suitable Blaschke products. The proof of the main result involves associating two piecewise analytic circle maps to the $(p,q,\infty)-$triangle group, mating these maps with appropriate Blaschke products to produce two commensurable conformal matings, and finally constructing the desired algebraic correspondence as a common lift of the two conformal matings.
We prove that commensurators of a geometrically rigid residually finite hyperbolic group have bounded average distortion.
Let $Y\to X$ be a proper map between proper hyperbolic metric spaces. A Cannon--Thurston map is a continuous extension $\partial Y \to \partial X$. We prove that in most known settings in which a Cannon--Thurston map exists it is uniformly finite-to-one. This answers a question due to Swarup from Bestvina's problem list and generalizes previous results of Cannon--Thurston, Kapovich--Lustig, Dowdall--Kapovich--Taylor and Ghosh.
We establish universality of the fat Basilica Julia set $J(z^2-\frac34)$ in conformal dynamics in the following sense: $J(z^2-\frac34)$ is quasiconformally equivalent to the fat Basilica Julia set of any polynomial as well as to the limit set of any geometrically finite closed surface Bers boundary group. We thus obtain the first example of a connected rational Julia set, not homeomorphic to the circle or the sphere, that is quasiconformally equivalent to a Kleinian limit set. It follows that any geometrically finite Bers boundary limit set is conformally removable. Other consequences of this universality result include quasi-symmetric uniformization of polynomial fat Basilicas by round Basilicas, and the existence of infinitely many non-commensurable uniformly quasi-symmetric surface subgroups of the Basilica quasi-symmetry group. We apply our techniques to cuspidal Basilica Julia sets arising from Schwarz reflections and cubic polynomials, yielding further universality classes. We also show that the standard Basilica Julia set $J(z^2-1)$ is the archbasilica in the David hierarchy.
In this paper, we develop techniques to study the Hausdorff dimensions of non-conical and Myrberg limit sets for groups acting on negatively curved spaces. We establish maximality of the Hausdorff dimension of the non-conical limit set of $G$ in the following cases. 1. $M$ is a finite volume complete Riemannian manifold of pinched negative curvature and $G$ is an infinite normal subgroups of infinite index in $\pi_1(M)$. 2. $G$ acts on a regular tree $X$ with $X/G$ infinite and amenable (dimension 1). 3. $G$ acts on the hyperbolic plane $\mathbb H^2$ such that $\mathbb H^2/G$ has Cheeger constant zero (dimension 2). 4. $G$ is a finitely generated geometrically infinite Kleinian group (dimension 3). We also show that the Hausdorff dimension of the Myrberg limit set is the same as the critical exponent, confirming a conjecture of Falk-Matsuzaki.
We develop an analog of the notion of a character variety in the context of algebraic correspondences. It turns out that matings of certain Fuchsian groups and polynomials are contained in this ambient character variety. This gives rise to two different analogs of the Bers slice by fixing either the polynomial or the Fuchsian group. The Bers-like slices are homeomorphic copies of Teichm\"uller spaces or combinatorial copies of polynomial connectedness loci. We show that these slices are bounded in the character variety, thus proving the analog of a theorem of Bers. To produce compactifications of the Bers-like slices, we initiate a study of degeneration of algebraic correspondences on trees of Riemann spheres, revealing a new degeneration phenomenon in conformal dynamics. There is no available analog of Sullivan's 'no invariant line field' theorem in our context. Nevertheless, for the four times punctured sphere, we show that the compactifications of Teichm\"uller spaces are naturally homeomorphic.
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.
We continue the study of the geometry of infinite geodesics in first passage percolation (FPP) on Gromov-hyperbolic groups G, initiated by Benjamini-Tessera and developed further by the authors. It was shown earlier by the authors that, given any fixed direction $\xi\in \partial G$, and under mild conditions on the passage time distribution, there exists almost surely a unique semi-infinite FPP geodesic from each $v\in G$ to $\xi$. Also, these geodesics coalesce to form a tree. Our main topic of study is the set of (random) exceptional directions for which uniqueness or coalescence fails. We study these directions in the context of two random geodesics trees: one formed by the union of all geodesics starting at a given base point, and the other formed by the union of all semi-infinite geodesics in a given direction $\xi\in \partial G$. We show that, under mild conditions, the set of exceptional directions almost surely has a strictly smaller Hausdorff dimension than the boundary, and hence has measure zero with respect to the Patterson-Sullivan measure. We also establish an upper bound on the maximum number of disjoint geodesics in the same direction. For groups that are not virtually free, we show that almost surely exceptional directions exist and are dense in $\partial G$. When the topological dimension of $\partial G$ is greater than one, we establish the existence of uncountably many exceptional directions. When the topological dimension of $\partial G$ is $n$, we prove the existence of directions $\xi$ with at least $(n+1)$ disjoint geodesics. Our results hinge on deep facts about hyperbolic groups. En route, we also establish facts about the structure of random bigeodesics that substantially strengthen prior results.
Motivated by Gromov's geodesic flow problem on hyperbolic groups $G$, we develop in this paper an analog using random walks. This leads to a notion of a harmonic analog $\Theta$ of the Bowen-Margulis-Sullivan measure on $\partial^2 G$. We provide three different but related constructions of $\Theta$: 1) by moving the base-point along a quasigeodesic ray 2) by moving the base-point along random walk trajectories 3) directly as a push-forward under the boundary map to $\partial^2 G$ of a measure inherited from studying all bi-infinite random walk trajectories (with no restriction on base-point) on $G^\mathbb{Z}$. Of these, the third construction is the most involved and needs new techniques. It relies on developing a framework where we can treat bi-infinite random walk trajectories as analogs of bi-infinite geodesics on complete simply connected negatively curved manifolds. Geodesic flow on a hyperbolic group is typically not well-defined due to non-uniqueness of geodesics. We circumvent this problem in the random walk setup by considering \emph{all} trajectories. We thus get a well-defined discrete flow that we call the \emph{randomized geodesic flow}, given by the $\mathbb{Z}-$shift on bi-infinite random walk trajectories. The $\mathbb{Z}-$shift is the random analog of the time one map of the geodesic flow. As an analog of ergodicity of the geodesic flow on a closed negatively curved manifold, we establish ergodicity of the $G$-action on $(\partial^2G, \Theta)$. As a consequence of our construction, we prove that the randomized geodesic flow is exponentially mixing of all orders and establish a functional CLT.
We prove a generalization of Bers' simultaneous uniformization theorem in the world of algebraic correspondences. More precisely, we construct algebraic correspondences that simultaneously uniformize a pair of non-homeomorphic genus zero orbifolds. We also present a complex-analytic realization of the Teichm\"uller space of a punctured sphere in the space of correspondences.
In this note we study a family of graphs of groups over arbitrary base graphs where all vertex groups are isomorphic to a fixed countable sofic group $G$, and all edge groups $H<G$ are such that the embeddings of $H$ into $G$ are identical everywhere. We prove soficity for this family of groups under a flexible technical hypothesis for $H$ called $\sigma$-co-sofic. This proves soficity for group doubles $*_H G$, where $H<G$ is an arbitrary separable subgroup and $G$ is countable and sofic. This includes arbitrary finite index group doubles of sofic groups among various other examples.
We produce a large class of hyperbolic homology 3-spheres admitting arbitrarily many distinct tight contact structures. We also produce a sub-class admitting arbitrarily many distinct tight contact structures within the same homotopy class of oriented plane distributions. As a corollary, we give a recipe to construct hyperbolic L-spaces admitting arbitrarily many distinct tight contact structures. We also introduce a notion of geometric limits of contact structures compatible with geometric limits of hyperbolic manifolds and study the behavior of the tight contact structures we construct under geometric limits.
We start with a Gromov-hyperbolic surface bundle $E$ over a graph, and drill out essential simple closed curves from fibers to obtain a drilled bundle $F$. We prove that for such drilled bundles $F$, the fundamental group $\pi_1(F)$ is relatively hyperbolic with $(\mathbb{Z}\oplus \mathbb{Z})$ peripheral groups. Combining the relative hyperbolicity of $\pi_1(F)$ thus obtained with a theorem of Wise, we establish virtually special cubulability of $\pi_1(F)$ provided that the maximal undrilled subbundles of $F$ are cubulable.
We develop a framework for common commensurators of discrete subgroups of lattices in isometry groups of CAT(0) spaces. We show that the Greenberg-Shalom hypothesis about discreteness of common commensurators of Zariski dense subgroups and lattices fails in this generality, even if one imposes strong finiteness conditions. We analyze some examples due to Burger and Mozes in this context and show that they have discrete common commensurator.
We show that if H is a non-elementary hyperbolic commensurated subgroup of infinite index in a hyperbolic group G, then H is virtually a free product of hyperbolic surface groups and free groups. We prove that whenever a one-ended hyperbolic group H is a fiber of a non-trivial hyperbolic bundle then H virtually splits over a 2-ended subgroup.
We discuss many surprising implications of a positive answer to a question raised in some cases by Greenberg in the $`70$s and more generally by Shalom in the early $2000$s. We refer to this positive answer as the Greenberg-Shalom hypothesis. This hypothesis then says that any infinite discrete subgroup of a semisimple Lie group with dense commensurator is a lattice in a product of some factors. For some applications it is natural to extend the hypothesis to cover semisimple algebraic groups over other fields as well.