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We prove that hyperbolic 2-bridge knots are determined amongst all compact 3-manifolds by the profinite completions of their knot groups., Comment: 8 pages

The purpose of Project LISTO is to support the instructional capacity of science educators and to validate innovative practices and strategies via previously developed interventions that address literacy-infused science and technology integration with standards-aligned curriculum. The evaluation of LISTO involved a multisite cluster randomized trial (CRT) designed to meet the Every Student Succeeds Act (ESSA) Tier 2 standards for "moderate" evidence, as well as the What Works Clearinghouse (WWC) standards "with reservations." The study estimated program impacts on both student and teacher outcomes and documented the fidelity of implementation and educators' perceptions of program quality. LISTO (Valid 45), and the corresponding VPD, VMC, and curricula resources did not lead to improved student achievement in science or reading for students consented to participate in the study. There was a negative impact on students' science achievement in each of the study years, except for Year 3, when science achievement data was not collected. These quantitative findings conflicted with qualitative data collected from LISTO teachers, who indicated that the program led to improvements in both science vocabulary and engagement and self-efficacy in science for students. LISTO teachers also indicated that the program had benefited their struggling readers, but there was no observed program impact on student reading achievement in any study year. While LISTO may have yielded some benefits for students, these benefits were not well captured on the standardized tests or survey instruments employed. LISTO had positive effects on teacher practices for a subsample of teachers, specifically on increased delivery of research-based instruction to teach science content as rated on a rubric by external reviewers. There were no differences in two other teacher outcomes, however, focused on the share of instructional time spent teaching new science content while performing various activities. The LISTO teachers who participated in the program reported that the Virtual Professional Development (VPD) and Virtual Mentoring and Coaching (VMC) were well-received by teachers. In sum, LISTO appeared to improve instructional practices for a sample of teachers who implemented the program for two years with complete data. Encouragingly, treatment teachers' overall positive reactions to the program suggest its potential to improve student affect and learning, but more extensive implementation experience by teachers and multi-year exposure by students starting in earlier grades may be needed to yield measurable benefits.

We show that in every commensurability class of cusped arithmetic hyperbolic manifolds of simplest type of dimension $2n+2\geq 6$ there are manifolds $M$ such that the Stiefel-Whitney classes $w_{2j}(M)$ are non-vanishing for all $0 \leq 2j \leq n$. We also show that for the same commensurability classes there are manifolds (different from the previous ones) that do not admit a $\text{spin}^\mathbb{C}$ structure., Comment: 13 pages. Added a new main result regarding manifolds without spin-c structures, and the corresponding secondary results. The authors thank Bruno Martelli for pointing out this application

We exhibit infinitely many pairs of non-isomorphic finitely presented, residually finite groups $\Delta$ and $\Gamma$ with $\Delta$ having Property FA, $\Gamma$ having a non-trivial action on a tree and $\Delta$ and $\Gamma$ having isomorphic profinite completions., Comment: 5 pages

We study how the spin structures on finite-volume hyperbolic n-manifolds restrict to cusps. When a cusp cross-section is a (n-1)-torus, there are essentially two possible behaviours: the spin structure is either bounding or Lie. We show that in every dimension n there are examples where at least one cusp is Lie, and in every dimension n <= 8 there are examples where all the cusps are bounding. By work of C. Bar, this implies that the spectrum of the Dirac operator is R in the first case, and discrete in the second. We therefore deduce that there are cusped hyperbolic manifolds whose spectrum of the Dirac operator is R in all dimensions, and whose spectrum is discrete in all dimensions n <= 8., Comment: 10 pages, 2 figures

Motivated by applications to pulling back quantum cellular automata from one manifold to another, we study the existence of immersions between certain smooth 4-manifolds. We show that they lead to a very interesting partial order on closed 4-manifolds., Comment: 43 pages, with an appendix by Alan W. Reid

The purpose of this study was to evaluate the Literacy-Infused Science Using Technology Innovation Opportunity (LISTO) validation project (Valid 45). LISTO was funded by the Investing in Innovation (i3) Fund and involved a multi-year intervention that provided virtual professional development and coaching, and literacy-infused science curricula to fifth-grade science teachers who taught predominantly low-income students and in predominantly rural public schools in Texas. The overarching goal of LISTO is to validate, via a 5-year longitudinal randomized controlled trial (RCT) study, literacy-infused science (LIS) instructional and curricular innovations to increase instructional capacity of teachers and to improve students' science and reading/writing literacy achievement in rural/non-rural schools for economically challenged (EC), inclusive of English language learners (ELL) students. Outcomes collected in the 2017-18 school year were considered to be exploratory, given the timing of Hurricane Harvey, which impacted Texas in August of 2017. Outcomes in the 2018-19 school year served as the confirmatory contrasts. LISTO resulted in increased teacher capacity to implement research-based strategies while teaching science content, yet this improvement did not necessarily translate into improved student achievement in science or reading. The LISTO professional development and coaching covered pedagogical strategies for teaching science, including those that have been shown to improve literacy and be particularly effective for ELs. There was a negative impact on students' science achievement in both 2017-18 (ES = -0.10) and in 2018--19 (ES = -0.13). There was a negative program impact on students' science interest (ES = -0.14), as measured by a survey, in 2017-18, and no impact in 2018-19. These quantitative findings were in conflict with qualitative data collected from LISTO teachers, who indicated that the program led to improvements in both science vocabulary and engagement and self-efficacy in science for students. LISTO had positive effects on teacher practices for a subsample of teachers, specifically on increased delivery of research-based instruction to teach science content as rated on a rubric by external reviewers (ES = +1.12). LISTO appeared to improve instructional practices for a sample of teachers who implemented the program for two years with complete data but did not positively impact student or teacher outcomes more broadly. However, results should be cautiously interpreted due to limitations of delayed and incomplete implementation in the first year of the project due to Hurricane Harvey. Encouragingly, teachers' overall positive reactions to the program suggest its potential to improve student affect and learning, but more extensive implementation experience by teachers and multi-year exposure by students starting from early grades may be needed to yield measurable benefits. Clearly, such focuses emerge as a highly recommended topic for future research.

In this paper we provide the first examples of arithmetic hyperbolic 3-manifolds that are rational homology spheres and bound geometrically either compact or cusped hyperbolic 4-manifolds., Comment: 19 pages, 1 figure, 1 table; improved exposition, some new results added; auxiliary SageMath code available at (https://github.com/sashakolpakov/dodecahedron-colouring)

In this paper we prove that certain hyperbolic link complements of $2$-tori in $S^4$ do not contain closed embedded totally geodesic hyperbolic $3$-manifolds., Comment: 18 pages. arXiv admin note: text overlap with arXiv:2005.11256

Let $\Gamma$ be a non-elementary Kleinian group and $H<\Gamma$ a finitely generated, proper subgroup. We prove that if $\Gamma$ has finite co-volume, then the profinite completions of $H$ and $\Gamma$ are not isomorphic. If $H$ has finite index in $\Gamma$, then there is a finite group onto which $H$ maps but $\Gamma$ does not. These results streamline the existing proofs that there exist full-sized groups that are profinitely rigid in the absolute sense. They build on a circleof ideas that can be used to distinguish among the profinite completions of subgroups of finite index in other contexts, e.g. limit groups. We construct new examples of profinitely rigid groups, including the fundamental group of the hyperbolic $3$-manifold ${\rm{Vol}}(3)$ and of the $4$-fold cyclic branched cover of the figure-eight knot. We also prove that if a lattice in ${\rm{PSL}}(2,\mathbb{C})$ is profinitely rigid, then so is its normalizer in ${\rm{PSL}}(2,\mathbb{C})$., Comment: Final version. To be published in a special issue of the Michigan Math. J. honoring Gopal Prasad

We introduce and study the notion of filling links in 3-manifolds: a link L is filling in M if for any 1-spine G of M which is disjoint from L, $\pi_1(G)$ injects into $\pi_1(M\smallsetminus L)$. A weaker "k-filling" version concerns injectivity modulo k-th term of the lower central series. For each k>1 we construct a k-filling link in the 3-torus. The proof relies on an extension of the Stallings theorem which may be of independent interest. We discuss notions related to "filling" links in 3-manifolds, and formulate several open problems. The appendix by C. Leininger and A. Reid establishes the existence of a filling hyperbolic link in any closed orientable 3-manifold with $\pi_1(M)$ of rank 2., Comment: 22 pages, 5 figures. v4: typos corrected

We prove that there are infinitely many non-homeomorphic hyperbolic knot complements $S^3\setminus K_i = \mathbb{H}^3/\Gamma_i$ for which $\Gamma_i$ contains elements whose trace is an algebraic non-integer., Comment: 15 pages, 6 figures; v2 includes answers to some questions in section 8 raised in v1

In this paper we study existence and lack thereof of closed embedded orientable co-dimension one totally geodesic submanifolds of minimal volume cusped orientable hyperbolic manifolds., Comment: 16 pages, 2 figure

We study embedding of closed totally geodesic hyperbolic 2-orbifolds in the Bianchi orbifolds $\mathbb{H}^3/PSL(2,\mathcal{O}_d)$. Our main result shows that there is a constant $c$ such that for $d$ large enough there are at least $cd$ closed embedded totally geodesic hyperbolic 2-orbifolds. Moreover we provide a list which conjecturally consists of those $d$ for which $\mathbb{H}^3/PSL(2,\mathcal{O}_d)$ does not contain a closed embedded totally geodesic hyperbolic 2-orbifold., Comment: 22 pages

This is a technical report accompanying the paper "All Principal Congruence Link Groups" (arXiv:1802.01275) classifying all principal congruence link complements in S^3 by the same authors. It provides a complete overview of all cases (d,I) that had to be considered, as well as describes the necessary computations and computer programs written for the classification result., Comment: 60 pages, 31 tables, 15 figures

This report lists the link diagrams in S^3 for all principal congruence link complements for which such a link diagram is known. Several unpublished link diagrams are included. Related to this, we also include one link diagram for an arithmetic regular tessellation link complement., Comment: 25 pages, 39 figures; Version 2 includes the new links found by Dunfield, Obeidin and Rudd

This paper describes in basic terms what a "Thin Group" is, as well as its uses in various subjects., Comment: 6 pages, 2 pictures

In this note, we show that there exist cusped hyperbolic $3$-manifolds that embed geodesically, but cannot bound geometrically. Thus, being a geometric boundary is a non-trivial property for such manifolds. Our result complements the work by Long and Reid on geometric boundaries of compact hyperbolic $4$-manifolds, and by Kolpakov, Reid and Slavich on embedding arithmetic hyperbolic manifolds., Comment: 11 pages, 3 figures; to appear in Proceedings AMS

We enumerate all the principal congruence link complements in $S^3$, there by answering a question of W. Thurston. Related articles: "Technical Report: All Principal Congruence Link Groups" (arXiv:1902.04722), "All Known Principal Congruence Links" (arXiv:1902.04426)., Comment: 7 pages, 1 table. V2: Addressing referee's comments, this article is now in announcement form, with technical details and link figures split off into separate articles

In this paper we provide a negative answer to a question of Farb about the relation between the algebraic degree of the stretch factor of a pseudo-Anosov homeomorphism and the genus of the surface on which it is defined., Comment: v2: Corrected minor errors

Let $M$ be a compact 3-manifold and $\Gamma=\pi_1(M)$. Work of Thurston and Culler--Shalen established the $\mathrm{SL}_2(\mathbb{C})$ character variety $X(\Gamma)$ as fundamental tool in the study of the geometry and topology of $M$. This is particularly the case when $M$ is the exterior of a hyperbolic knot $K$ in $S^3$. The main goals of this paper are to bring to bear tools from algebraic and arithmetic geometry to understand algebraic and number theoretic properties of the so-called canonical component of $X(\Gamma)$, as well as distinguished points on the canonical component, when $\Gamma$ is a knot group. In particular, we study how the theory of quaternion Azumaya algebras can be used to obtain algebraic and arithmetic information about Dehn surgeries, and perhaps of most interest, to construct new knot invariants that lie in the Brauer groups of curves over number fields., Comment: v2 Revisions and corrections; v3 To appear in IMRN