arXiv subjects
Publications and source records attributed to D. B. McReynolds.
We show that all closed geodesics in any arithmetic complex hyperbolic surface of second type are simple. These include the fake projective planes studied by Mumford, Klingler, Prasad-Yeung, and Cartwright-Steger.
We investigate the homological dimensions of discrete subgroups of non-compact simple Lie groups using a flow recently introduced by the authors. Using new estimates on the critical index for a broad class of discrete subgroups, we prove that the homological dimension of non-lattice, discrete, Zariski-dense subgroups is bounded above by $n-αr$ where $n$ is the dimension of the associated symmetric space, $r$ is the real rank, and $α\geq \frac{1}{8}$. Under the assumption that the discrete group has regular limit cone, our bound improves to $\frac{5}{6}n+1$. Additionally, we make some conjectures on the possible homological dimensions for non-lattice discrete subgroups and pose a few questions in the more general semisimple setting.
We show that for non-conjugate subgroups $G_1$ and $G_2$ of a finite group $G$ there exists an extension of $G$ (by a finite group) in which the pre-images of $G_1$ and $G_2$ are not isomorphic. This allows us to show that $\mathbb Z$-coset equivalent subgroups of a finite group are not necessarily isomorphic, answering a question of Dipendra Prasad. We also indicate connections to profinite rigidity, anabelian geometry, mapping class groups, and non-arithmetic lattices in Lie groups.
Inspired by work of Besson-Courtois-Gallot, we construct a flow called the natural flow on a non-positively curved Riemannian manifold $M$. As with the natural map, the $k$-Jacobian of the natural flow is directly related to the critical exponent $δ$ of the fundamental group. There are several applications of the natural flow that connect dynamical, geometrical, and topological invariants of the manifold. First, we give $k$-dimensional linear isoperimetric inequalities when $k > δ$. This, in turn, produces lower bounds on the Cheeger constant. We resolve a recent conjecture of Dey-Kapovich on the non-existence of $k$-dimensional compact, complex subvarieties of complex hyperbolic manifolds with $2k > δ$. We also provide upper bounds on the homological dimension, generalizing work of Kapovich and work of Farb with the first two authors. Using the natural flow together with Morse theory, we also give upper bounds on the cohomological dimension, which partially resolve a conjecture of Kapovich. Finally, we introduce a new growth condition on the Bowen-Margulis measure that we call uniformly exponentially bounded that we connect to the cohomological dimension and which could be of independent interest.
We establish an analog of a theorem of Stallings which asserts the homomorphisms between the universal nilpotent quotients induced by a homomorphism $G \to H$ of groups are isomorphisms provided a pair of homological conditions are satisfied. Our analogy does not have a homomorphism between $G$ and $H$ but instead $G,H \leq G_0$ that satisfies a similar homological condition. We derive a few applications of this result. First, we show that there exist pairs of non-isomorphic number fields whose absolute Galois groups have isomorphic universal nilpotent quotients. We show that there exists pairs of non-isometric hyperbolic $n$-manifolds whose fundamental groups are residually nilpotent and have isomorphic universal nilpotent quotients. These are the first examples of residually nilpotent Kleinian groups with arbitrarily large nilpotent genus. Complex hyperbolic 2-manifold examples are given as well. Considering Riemann surfaces and complex hyperbolic 2-manifolds as projective curves and surfaces defined over a number field, we show the (outer) action of the absolute Galois group of the field of definition on the universal nilpotent quotients of the geometric fundamental groups are equivalent. This is in contrast to fact that the (outer) Galois action on the geometric fundamental group of a projective hyperbolic curve determines the curve by work of Mochizuki. In particular, the nilpotent representation theory of the geometric fundamental group is not anabelian.
We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group $\mathrm{PSL}(2,\mathbb{Z}[ω])$ with $ω^2+ω+1=0$ is rigid in this sense. Other examples include the non-uniform lattice of minimal co-volume in $\mathrm{PSL}(2,\mathbb{C})$ and the fundamental group of the Weeks manifold (the closed hyperbolic $3$-manifold of minimal volume).
We prove that certain Fuchsian triangle groups are profinitely rigid in the absolute sense, i.e. each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. We also develop a method based on character varieties that can be used to distinguish between the profinite completions of certain groups.
For finitely generated groups $G$ and $H$ equipped with word metrics, a translation-like action of $H$ on $G$ is a free action where each element of $H$ moves elements of $G$ a bounded distance. Translation-like actions provide a geometric generalization of subgroup containment. Extending work of Cohen, we show that cocompact lattices in a general semisimple Lie group $\mathbf{G}$ that is not isogenous to $\mathrm{SL}(2,\mathbb{R})$ admit translation-like actions by $\mathbb{Z}^2$. This result follows from a more general result. Namely, we prove that any cocompact lattice in the unipotent radical $\mathbf{N}$ of the Borel subgroup $\mathbf{AN}$ of $\mathbf{G}$ acts translation-like on any cocompact lattice in $\mathbf{G}$. We also prove that for noncompact simple Lie groups $G,H$ with $H<G$ and lattices $Γ< G$ and $Δ< H$, that $Γ/Δ$ is quasi-isometric to $G/H$ where $Γ/Δ$ is the quotient via a translation-like action of $Δ$ on $Γ$.
We study closed non-positively curved Riemannian manifolds $M$ which admit `fat $k$-flats': that is, the universal cover $\tilde M$ contains a positive radius neighborhood of a $k$-flat on which the sectional curvatures are identically zero. We investigate how the fat $k$-flats affect the cardinality of the collection of closed geodesics. Our first main result is to construct rank $1$ non-positively curved manifolds with a fat $1$-flat which corresponds to a twisted cylindrical neighborhood of a geodesic on $M$. As a result, $M$ contains an embedded closed geodesic with a flat neighborhood, but $M$ nevertheless has only countably many closed geodesics. Such metrics can be constructed on finite covers of arbitrary odd-dimensional finite volume hyperbolic manifolds. Our second main result is to prove a closing theorem for fat flats, which implies that a manifold $M$ with a fat $k$-flat contains an immersed, totally geodesic $k$-dimensional flat closed submanifold. This guarantees the existence of uncountably many closed geodesics when $k \geq 2$. Finally, we collect results on thermodynamic formalism for the class of manifolds considered in this paper.
In this article, we investigate when the set of primitive geodesic lengths on a Riemannian manifold have arbitrarily long arithmetic progressions. We prove that in the space of negatively curved metrics, a metric having such arithmetic progressions is quite rare. We introduce almost arithmetic progressions, a coarsification of arithmetic progressions, and prove that every negatively curved, closed Riemannian manifold has arbitrarily long almost arithmetic progressions in its primitive length spectrum. Concerning genuine arithmetic progressions, we prove that every non-compact, locally symmetric, arithmetic manifold has arbitrarily long arithmetic progressions in its primitive length spectrum. We end with a conjectural characterization of arithmeticity in terms of arithmetic progressions in the primitive length spectrum. We also suggest an approach to a well known spectral rigidity problem based on the scarcity of manifolds with arithmetic progressions.
In this article, we prove that the commensurability class of a closed, orientable, hyperbolic 3-manifold is determined by the surface subgroups of its fundamental group. Moreover, we prove that there can be only finitely many closed, orientable, hyperbolic 3-manifolds that have the same set of surfaces.
In this paper we construct arbitrarily large families of smooth projective varieties and closed Riemannian manifolds that share many algebraic and analytic invariants. For instance, every non-arithmetic, closed hyperbolic $3$--manifold admits arbitrarily large collections of non-isometric finite covers which are strongly isospectral, length isospectral, and have isomorphic integral cohomology where the isomorphisms commute with restriction and co-restriction. We can also construct arbitrarily large collections of pairwise non-isomorphic smooth projective surfaces where these isomorphisms in cohomology are natural with respect to Hodge structure or as Galois modules. In particular, the projective varieties have isomorphic Picard and Albanese varieties, and they also have isomorphic effective Chow motives. Our construction employs an integral refinement of the Gassman--Sunada construction that has recently been utilized by D. Prasad. One application of our work shows the non-injectivity of the map from the Grothendieck group of varieties over $\overline{\mathbf{Q}}$ to the Grothendieck group of the category of effective Chow motives. We also answer a question of D. Prasad.
The purpose of this article is to produce effective versions of some rigidity results in algebra and geometry. On the geometric side, we focus on the spectrum of primitive geodesic lengths (resp., complex lengths) for arithmetic hyperbolic 2-manifolds (resp., 3-manifolds). By work of Reid, this spectrum determines the commensurability class of the 2-manifold (resp., 3-manifold). We establish effective versions of these rigidity results by ensuring that, for two incommensurable arithmetic manifolds of bounded volume, the length sets (resp., the complex length sets) must disagree for a length that can be explicitly bounded as a function of volume. We also prove an effective version of a similar rigidity result established by the second author with Reid on a surface analog of the length spectrum for hyperbolic 3-manifolds. These effective results have corresponding algebraic analogs involving maximal subfields and quaternion subalgebras of quaternion algebras. To prove these effective rigidity results, we establish results on the asymptotic behavior of certain algebraic and geometric counting functions which are of independent interest.
The geodesic length spectrum of a complete, finite volume, hyperbolic 3-orbifold M is a fundamental invariant of the topology of M via Mostow-Prasad Rigidity. Motivated by this, the second author and Reid defined a two-dimensional analogue of the geodesic length spectrum given by the multiset of isometry types of totally geodesic, immersed, finite-area surfaces of M called the geometric genus spectrum. They showed that if $M$ is arithmetic and contains a totally geodesic surface, then the geometric genus spectrum of M determines its commensurability class. In this paper we define a coarser invariant called the totally geodesic area set given by the set of areas of surfaces in the geometric genus spectrum. We prove a number of results quantifying the extent to which non-commensurable arithmetic hyperbolic 3-orbifolds can have arbitrarily large overlaps in their totally geodesic area sets.
In 1992, Reid asked whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building towards a negative answer to this question, Futer and Millichap recently constructed infinitely many pairs of non-commensurable, non-arithmetic hyperbolic 3-manifolds which have the same volume and whose length spectra begin with the same first $m$ geodesic lengths. In the present paper, we show that this phenomenon is surprisingly common in the arithmetic setting. In particular, given any arithmetic hyperbolic 3-orbifold derived from a quaternion algebra, any finite subset $S$ of its geodesic length spectrum, and any $k \geq 2$, we produce infinitely many $k$-tuples of arithmetic hyperbolic 3-orbifolds which are pairwise non-commensurable, have geodesic length spectra containing $S$, and have volumes lying in an interval of (universally) bounded length. The main technical ingredient in our proof is a bounded gaps result for prime ideals in number fields lying in Chebotarev sets which extends recent work of Thorner.
Our main result is that for all sufficiently large $x_0>0$, the set of commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with fixed invariant trace field $k$ and systole bounded below by $x_0$ has density one within the set of all commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with invariant trace field $k$. The proof relies upon bounds for the absolute logarithmic Weil height of algebraic integers due to Silverman, Brindza and Hajdu, as well as precise estimates for the number of rational quaternion algebras not admitting embeddings of any quadratic field having small discriminant. When the trace field is $\mathbf{Q}$, using work of Granville and Soundararajan, we establish a stronger result that allows our constant lower bound $x_0$ to grow with the area. As an application, we establish a systolic bound for arithmetic hyperbolic surfaces that is related to prior work of Buser-Sarnak and Katz-Schaps-Vishne. Finally, we establish an analogous density result for commensurability classes of arithmetic hyperbolic 3-orbifolds with small area totally geodesic $2$-orbifolds.
In this paper, we construct families of nonisometric hyperbolic orbifolds that contain the same isometry classes of nonflat totally geodesic subspaces. The main tool is a variant of the well-known Sunada method for constructing length-isospectral Riemannian manifolds that handles totally geodesic submanifolds of multiple codimensions simultaneously.
Given a pair of number fields with isomorphic rings of adeles, we construct bijections between objects associated to the pair. For instance we construct an isomorphism of Brauer groups that commutes with restriction. We additionally construct bijections between central simple algebras, maximal orders, various Galois cohomology sets, and commensurability classes of arithmetic lattices in simple, inner algebraic groups. We show that under certain conditions, lattices corresponding to one another under our bijections have the same covolume and pro-congruence completion. We also make effective a finiteness result of Prasad and Rapinchuk.