arXiv subjects
Publications and source records attributed to Ian Biringer.
We study the homeomorphism types of certain covers of (always orientable) surfaces, usually of infinite-type. We show that every surface with non-abelian fundamental group is covered by every noncompact surface, we identify the universal abelian covers and the $\mathbb{Z}/n\mathbb{Z}$-homology covers of surfaces, and we show that non-locally finite characteristic covers of surfaces have four possible homeomorphism types.
We prove a finiteness theorem for subgroups of bounded rank in hyperbolic $3$-manifold groups. As a consequence, we show that every bounded rank covering tower of closed hyperbolic $3$-manifolds is a tower of finite covers associated to a fibration over a $1$-orbifold.
We show that except for one exceptional case, a lamination on the boundary of a 3-dimensional handlebody H is a Hausdorff limit of meridians if and only if it is commensurable to a lamination with a 'homoclinic leaf'. This is a precise version of a philosophy called Casson's Criterion, which appeared in unpublished notes of A. Casson. Applications include a characterization of when a non-minimal lamination is a Hausdorff limit of meridians, in terms of properties of its minimal components, and a related characterization of which reducible self-homeomorphisms of the boundary of H have powers that extend to subcompressionbodies of H.
We construct a geometric decomposition for the convex core of a thick hyperbolic 3-manifold M with bounded rank. Corollaries include upper bounds in terms of rank and injectivity radius on the Heegaard genus of M and on the radius of any embedded ball in the convex core of M.
We study unimodular measures on the space $\mathcal M^d$ of all pointed Riemannian $d$-manifolds. Examples can be constructed from finite volume manifolds, from measured foliations with Riemannian leaves, and from invariant random subgroups of Lie groups. Unimodularity is preserved under weak* limits, and under certain geometric constraints (e.g. bounded geometry) unimodular measures can be used to compactify sets of finite volume manifolds. One can then understand the geometry of manifolds $M$ with large, finite volume by passing to unimodular limits. We develop a structure theory for unimodular measures on $\mathcal M^d$, characterizing them via invariance under a certain geodesic flow, and showing that they correspond to transverse measures on a foliated `desingularization' of $\mathcal M^d$. We also give a geometric proof of a compactness theorem for unimodular measures on the space of pointed manifolds with pinched negative curvature, and characterize unimodular measures supported on hyperbolic $3$-manifolds with finitely generated fundamental group.
This is the first of two papers on the global topology of the space $\textrm{Sub}(G)$ of all closed subgroups of $G=\textrm{PSL}_2(\mathbb{R})$, equipped with the Chabauty topology. In this paper, we study the spaces of lattices and elementary subgroups of $G$, and prove a continuity result for conformal grafting of (possibly infinite type) vectored orbifolds that will be useful in both papers. More specifically, we first identify the homotopy type of the space of elementary subgroups of $G$, following Baik-Clavier. Then for a fixed finite type hyperbolizable $2$-orbifold $S$, we show that the space $\textrm{Sub}_S(G)$ of all lattices $Γ< G$ with $Γ\backslash \mathbb{H}^2 \cong S$ is a fiber orbibundle over the moduli space $\mathcal M(S)$. We describe the closure $\overline{\textrm{Sub}_S(G)}$ in $\textrm{Sub}(G)$ and show that $\partial \textrm{Sub}_S(G)$ has a neighborhood deformation retract within $\overline{\textrm{Sub}_S(G)}$. When $S$ is not one of finitely many low complexity orbifolds, we show that $\overline{\textrm{Sub}_S(G)}$ is simply connected. In the simplest exceptional case, when $S$ is a sphere with three total cusps and cone points, we show that $\overline{\textrm{Sub}_S(G)}$ is a (usually nontrivial) lens space. Finally, we show that when $(X_i,v_i) \to (X_\infty,v_\infty)$ is a (possibly infinite type) smoothly converging sequence of vectored hyperbolic $2$-orbifolds, and we graft in Euclidean annuli along suitable collections of simple closed curves in the $X_i$, then after uniformization, the resulting vectored hyperbolic $2$-orbifolds converge smoothly to the expected limit. As part of the proof, we give a new lower bound on the hyperbolic distance between points in a grafted orbifold in terms of their original distance.
We study the convergence of volume-normalized Betti numbers in Benjamini-Schramm convergent sequences of non-positively curved manifolds with finite volume. In particular, we show that if $X$ is an irreducible symmetric space of noncompact type, $X \neq \mathbb H^3$, and $(M_n)$ is any Benjamini-Schramm convergent sequence of finite volume $X$-manifolds, then the normalized Betti numbers $b_k(M_n)/vol(M_n)$ converge for all $k$. As a corollary, if $X$ has higher rank and $(M_n)$ is any sequence of distinct, finite volume $X$-manifolds, the normalized Betti numbers of $M_n$ converge to the $L^2$ Betti numbers of $X$. This extends our earlier work with Nikolov, Raimbault and Samet, where we proved the same convergence result for uniformly thick sequences of compact $X$-manifolds.
We study invariant random subgroups (IRSs) of semidirect products $G = A \rtimes Γ$. In particular, we characterize all IRSs of parabolic subgroups of $\mathrm{SL}_d(\mathbb{R})$, and show that all ergodic IRSs of $\mathbb{R}^d \rtimes \mathrm{SL}_d(\mathbb{R})$ are either of the form $\mathbb{R}^d \rtimes K$ for some IRS of $\mathrm{SL}_d(\mathbb{R})$, or are induced from IRSs of $Λ\rtimes \mathrm{SL}(Λ)$, where $Λ< \mathbb{R}^d$ is a lattice.
We show that if the monodromy of a 3-manifold M that fibers over the circle has large translation distance in the curve complex, then the rank of the fundamental group of M is 2g+1, where g is the genus of the fiber.
We describe an explicit metric that induces the Chabauty topology on the space of closed subsets of a proper metric space M.
We use the geometry of the Farey graph to give an alternative proof of the fact that if $A \in GL_2\mathbb Z$ and $G_A=\mathbb Z^2 \rtimes_A \mathbb Z$ is generated by two elements, there is a single Nielsen equivalence class of $2$-element generating sets for $G_A$ unless $A$ is conjugate to $\pm \left(\begin {smallmatrix} 2 & 1 \\ 1 & 1 \end {smallmatrix}\right )$, in which case there are two.
We prove that there is an algorithm to determine if a given finite graph is an induced subgraph of a given curve graph.
We study the asymptotic behaviour of Betti numbers, twisted torsion and other spectral invariants of sequences of locally symmetric spaces. Our main results are uniform versions of the DeGeorge--Wallach Theorem, of a theorem of Delorme and various other limit multiplicity theorems. A basic idea is to adapt the notion of Benjamini--Schramm convergence (BS-convergence), originally introduced for sequences of finite graphs of bounded degree, to sequences of Riemannian manifolds, and analyze the possible limits. We show that BS-convergence of locally symmetric spaces implies convergence, in an appropriate sense, of the associated normalized relative Plancherel measures. This then yields convergence of normalized multiplicities of unitary representations, Betti numbers and other spectral invariants. On the other hand, when the corresponding Lie group $G$ is simple and of real rank at least two, we prove that there is only one possible BS-limit, i.e. when the volume tends to infinity, locally symmetric spaces always BS-converge to their universal cover $G/K$. This leads to various general uniform results. When restricting to arbitrary sequences of congruence covers of a fixed arithmetic manifold we prove a strong quantitative version of BS-convergence which in turn implies upper estimates on the rate of convergence of normalized Betti numbers in the spirit of Sarnak--Xue. An important role in our approach is played by the notion of Invariant Random Subgroups. For higher rank simple Lie groups $G$, we exploit rigidity theory, and in particular the Nevo--Stück--Zimmer theorem and Kazhdan's property (T), to obtain a complete understanding of the space of IRSs of $G$.
In the first paper of this series (arxiv.org/abs/1210.2961) we studied the asymptotic behavior of Betti numbers, twisted torsion and other spectral invariants for sequences of lattices in Lie groups G. A key element of our work was the study of invariant random subgroups (IRSs) of G. Any sequence of lattices has a subsequence converging to an IRS, and when G has higher rank, the Nevo-Stuck-Zimmer theorem classifies all IRSs of G. Using the classification, one can deduce asymptotic statments about spectral invariants of lattices. When G has real rank one, the space of IRSs is more complicated. We construct here several uncountable families of IRSs in the groups SO(n,1). We give dimension-specific constructions when n=2,3, and also describe a general gluing construction that works for every n at least 2. Part of the latter construction is inspired by Gromov and Piatetski-Shapiro's construction of non-arithmetic lattices in SO(n,1).
Przytycki has shown that the size $\mathcal{N}_{k}(S)$ of a maximal collection of simple closed curves that pairwise intersect at most $k$ times on a topological surface $S$ grows at most as a polynomial in $|χ(S)|$ of degree $k^{2}+k+1$. In this paper, we narrow Przytycki's bounds by showing that $$ \mathcal{N}_{k}(S) =O \left( \frac{ |χ|^{3k}}{ ( \log |χ| )^2 } \right) , $$ In particular, the size of a maximal 1-system grows sub-cubically in $|χ(S)|$. The proof uses a circle packing argument of Aougab-Souto and a bound for the number of curves of length at most $L$ on a hyperbolic surface. When the genus $g$ is fixed and the number of punctures $n$ grows, we can improve our estimates using a different argument to give $$ \mathcal{N}_{k}(S) \leq O(n^{2k+2}) . $$ Using similar techniques, we also obtain the sharp estimate $\mathcal{N}_{2}(S)=Θ(n^3)$ when $k=2$ and $g$ is fixed.
We study the ends of a generic manifold, with respect to a unimodular measure on the space of pointed Riemannian manifolds with bounded curvatures. We apply our general result to the case of surfaces and obtain as corollaries a very precise description of generic leaves for foliations with invariant measures and of quotients of the hyperbolic plane by invariant random subgroups of the isometry group.
When $S$ is a closed, orientable surface with genus $g(S) \geq 2$, we show that the automorphism group of the compression body graph $\mathcal{CB}(S)$ is the mapping class group. Here, vertices are compression bodies with exterior boundary $S$, and edges connect pairs of compression bodies where one contains the other.
An invariant random subgroup $H \leq G$ is a random closed subgroup whose law is invariant to conjugation by all elements of $G$. When $G$ is locally compact and second countable, we show that for every invariant random subgroup $H \leq G$ there almost surely exists an invariant measure on $G/H$. Equivalently, the modular function of $H$ is almost surely equal to the modular function of $G$, restricted to $H$. We use this result to construct invariant measures on orbit equivalence relations of measure preserving actions. Additionally, we prove a mass transport principle for discrete or compact invariant random subgroups.