arXiv subjects
Publications and source records attributed to Andrew Zimmer.
We obtain orbital counting results for relatively Anosov groups with respect to linear functionals with finite critical exponent. Our counting results follow from an equidistribution result and rely crucially on previous equidistribution results obtained in our proof of counting results for periods. Our results generalize earlier work of Sambarino in the setting of Anosov groups.
Let $\Gamma$ be a discrete subgroup of a connected semisimple real algebraic group of higher rank. The growth indicator function $\psi_\Gamma$ records the directional exponential growth of the Cartan projections of elements of $\Gamma$ in the positive Weyl chamber $\mathfrak a^+$. We prove that if $\Gamma$ is a non-elementary relatively Borel Anosov group, then $\psi_\Gamma$ is strictly concave on non-collinear directions. We prove this by establishing the $\mathcal C^1$-smoothness of the Manhattan hypersurface, defined as the unit level set of the critical-exponent map $\phi\mapsto\delta^\phi(\Gamma)$. More generally, for a non-elementary $\theta$-transverse group, we prove local $\mathcal C^1$-regularity near every point of the $\theta$-Manhattan hypersurface that is positive on the $\theta$-limit cone and has a critical gap at infinity. In particular, the $\theta$-Manhattan hypersurface is globally $\mathcal C^1$ for relatively $\theta$-Anosov groups, and their $\theta$-growth indicator functions are strictly concave on non-collinear directions.
In higher rank, there is a well-studied theory of Patterson--Sullivan measures supported on partial flag manifolds. However, establishing the existence and uniqueness of such measures is a difficult question. In this paper, we develop a theory for Patterson--Sullivan measures supported on certain vector-valued horofunction boundaries of the associated symmetric space, where existence is straightforward. We also introduce a notion of shadows for this compactification and establish a shadow lemma. For transverse groups, we prove uniqueness and ergodicity results.
In this paper we develop a theory for Patterson--Sullivan measures for non-Borel Anosov groups on the Furstenberg boundary. Previously, such a theory has been successfully developed for measures supported on the partial flag manifold associated to the Anosov condition, which coincides with the Furstenberg boundary only under the strongest Anosov condition, Borel Anosov. We establish existence, uniqueness, and ergodicity results for the measures on the Furstenberg boundary under arbitrary Anosov conditions; we show ergodicity of Bowen--Margulis--Sullivan measures on the homogeneous space; and we establish strict convexity results for the critical exponent associated to functionals on the entire Cartan subspace. Using this strict convexity, we establish an entropy rigidity result for Anosov groups with Lipschitz limit set. A key tool we develop is a new sufficient condition for the existence of a measurable boundary map associated to a Zariski dense representation. This result not only applies to Anosov groups, but also to transverse groups, mapping class groups, and discrete subgroups of the isometry groups of Gromov hyperbolic spaces.
In this paper, we prove a rigidity result for proper holomorphic maps between unit balls that have many symmetries and which extend to H\"older continuous maps on the boundary, with H\"older exponent strictly greater than 1/2.
In this paper, we consider random walks on the isometry groups of general metric spaces. Under some mild conditions, we show that if two non-elementary random walks on a discrete subgroup of the isometry group have non-singular stationary measures, then subgroups generated by the random walks are commensurable. This result in particular applies to Gromov hyperbolic spaces and Teichm\"uller spaces. As a specific application, we prove singularity between stationary measures associated to random walks on different fiber subgroups of the fundamental group of a hyperbolic 3-manifold fibering over the circle.
Farre, Pozzetti and Viaggi proved that any (d-k)-hyperconvex subgroup of PSL(d,C) is virtually isomorphic to a convex cocompact Kleinian group and that its k-th simple root critical exponent is at most 2. We show that a (d-k)-hyperconvex subgroup is isomorphic to a uniform lattice in PSL(2,C) if and only if its k-th simple root critical exponent is exactly 2. Furthermore, we show that if a strongly irreducible (d-k)-hyperconvex subgroup has k-th simple root critical exponent 2, then it is the image of a uniform lattice in PSL(2, C) by an irreducible representation of PSL(2, C) into PSL(d, C).
We prove a logarithm law-type result for the spiraling of geodesics around certain types of compact subsets (e.g. quotients of periodic Morse flats) in quotients of rank one CAT(0) spaces.
We consider random walks on semisimple Lie groups where the support of the step distribution generates (as a group) a Zariski dense discrete subgroup of infinite covolume. When the semisimple Lie group has property (T), we show that the stationary measure on the Furstenberg boundary is singular to the Lebesgue measure class. This result does not require any condition on the moment or symmetry of the step distribution. When the semisimple Lie group has rank one and the step distribution has a finite first moment, we again show that the stationary measure on the Furstenberg boundary is singular to the Lebesgue measure class. For general semisimple Lie groups, we also obtain a sufficient condition for the singularity of the stationary measure and a general Patterson-Sullivan measure.
In this paper we introduce Patterson--Sullivan systems, which consist of a group action on a compact metrizable space and a quasi-invariant measure which behaves like a classical Patterson--Sullivan measure. For such systems we prove a generalization of Tukia's measurable boundary rigidity theorem. We then apply this generalization to (1) study the singularity conjecture for Patterson--Sullivan measures (or, conformal densities) and stationary measures of random walks on isometry groups of Gromov hyperbolic spaces, mapping class groups, and discrete subgroups of semisimple Lie groups; (2) prove versions of Tukia's theorem for word hyperbolic groups, Teichm\"uller spaces, and higher rank symmetric spaces; and (3) in a companion paper prove an entropy rigidity result for Anosov groups with Lipschitz limit sets.
We introduce a class of complex manifolds which we call weakly holomorphic homogeneous regular manifolds (wHHR) manifolds. As the name suggests, this class contains the so-called holomorphic homogeneous regular manifolds but also other classes of complex manifolds such as two dimensional finite type domains and simply connected K\"ahler manifolds with pinched negative sectional curvature. For wHHR Stein manifolds we prove that the Bergman and Kobayashi metrics are biLipschitz equivalent.
We survey recent work on the geometry and dynamics of transverse subgroups of semi-simple Lie groups.
In this paper we investigate the Gromov hyperbolicity of the classical Kobayashi and Hilbert metrics, and the recently introduced minimal metric. Using the linear isoperimetric inequality characterization of Gromov hyperbolicity, we show if these metrics have an "expanding property" near the boundary, then they are Gromov hyperbolic. This provides a new characterization of the convex domains whose Hilbert metric is Gromov hyperbolic, a new proof of Balogh-Bonk's result that the Kobayashi metric is Gromov hyperbolic on a strongly pseudoconvex domain, a new proof of the second author's result that the Kobayashi metric is Gromov hyperbolic on a convex domain with finite type, and a new proof of Fiacchi's result that the minimal metric is Gromov hyperbolic on a strongly minimally convex domain. We also characterize the smoothly bounded convex domains where the minimal metric is Gromov hyperbolic.
We obtain a bi-Lipschitz rigidity theorem for a Zariski dense discrete subgroup of a connected simple real algebraic group. As an application, we show that any Zariski dense discrete subgroup of a higher rank semisimple algebraic group $G$ cannot have a $C^1$-smooth slim limit set in $G/P$ for any non-maximal parabolic subgroup $P$.
In this paper we develop a theory of Patterson--Sullivan measures associated to coarse cocycles of convergence groups. This framework includes Patterson-Sullivan measures associated to the Busemann cocycle on the geodesic boundary of a Gromov hyperbolic metric spaces and Patterson-Sullivan measures on flag manifolds associated to Anosov (or more general transverse) subgroups of semisimple Lie groups, as well as more examples. Under some natural geometric assumptions on the coarse cocycle, we prove existence, uniqueness, and ergodicity results.
We establish counting, mixing and equidistribution results for finite BMS measures on flow spaces associated to geometrically finite convergence group actions. We show that, in particular, these results apply to flow spaces associated to relatively Anosov groups.
We establish a smoothness result for families of biholomorphisms between smooth families of strongly pseudoconvex domains, each with trivial biholomorphism group. This is accomplished by considering the Riemannian geometry of their Bergman metrics and proving a result about the smoothness of families of isometries between smooth families of Riemannian manifolds.
We establish existence, uniqueness and ergodicity results for Patterson-Sullivan measures for relatively Anosov groups. As applications we obtain an entropy gap theorem and a strict concavity result for entropies associated to linear functionals.