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A group is LERF (locally extended residually finite) if all its finitely generated subgroups are separable. We prove that the trialitarian arithmetic lattices in $\mathbf{PSO}_{7,1}(\mathbb{R})$ are not LERF. This result, together with previous work by the third author, implies that all arithmetic lattices in $\mathbf{PO}_{n,1}(\mathbb{R})$, $n>3$, are not LERF., Comment: 8 pages, 1 figure (minor changes like typos, etc)

This paper shows that immersed totally geodesic $m$-dimensional suborbifolds of $n$-dimensional arithmetic hyperbolic orbifolds correspond to finite subgroups of the commensurator whenever $m \geqslant \lfloor \frac{n}{2} \rfloor$. We call such totally geodesic suborbifolds finite centraliser subspaces (or fc-subspaces) and use them to formulate an arithmeticity criterion for hyperbolic lattices. We show that a hyperbolic orbifold $M$ is arithmetic if and only if it has infinitely many fc-subspaces, exhibiting examples of non-arithmetic orbifolds that contain non-fc subspaces of codimension one. We provide an algebraic characterisation of totally geodesically immersed suborbifolds of arithmetic hyperbolic orbifolds by analysing Vinberg's commensurability invariants. This allows us to construct examples with the property that the adjoint trace field of the geodesic suborbifold properly contains the adjoint trace field of the orbifold. The case of special interest is that of exceptional trialitarian $7$-dimensional orbifolds. We show that every such orbifold contains a totally geodesic arithmetic hyperbolic $3$-orbifold of exceptional type. Finally, we study arithmetic properties of orbifolds that descend to their totally geodesic suborbifolds, proving that all suborbifolds in a (quasi-)arithmetic orbifold are (quasi-)arithmetic., Comment: 71 pages (with new margins), 4 figures, substantial update from v1, added table of contents, improved exposition, typos corrected; + Section 3.4: new results about trialitarian lattices. Major revision of Section 5. Added preliminary Section 2.1; + minor revision

In the present paper, we construct a cusped hyperbolic $4$-manifold with all cusp sections homeomorphic to the Hantzsche-Wendt manifold, which is a rational homology sphere. By a result of Gol\'enia and Moroianu, the Laplacian on $2$-forms on such a manifold has purely discrete spectrum. This shows that one of the main results of Mazzeo and Phillips from 1990 cannot hold without additional assumptions on the homology of the cusps. This also answers a question by Gol\'enia and Moroianu from 2012. We also correct and refine the incomplete classification of compact orientable flat $3$-manifolds arising from cube colourings provided earlier by the last two authors., Comment: 15 pages, 1 figure, 1 table; SageMath worksheets available at https://github.com/sashakolpakov/24-cell-colouring

We show that every plumbing of disc bundles over surfaces whose genera satisfy a simple inequality may be embedded as a convex submanifold in some closed hyperbolic four-manifold. In particular its interior has a geometrically finite hyperbolic structure that covers a closed hyperbolic four-manifold., Comment: 18 pages, 11 figures

This paper shows that many hyperbolic manifolds obtained by glueing arithmetic pieces embed into higher-dimensional hyperbolic manifolds as codimension-one totally geodesic submanifolds. As a consequence, many Gromov--Pyatetski-Shapiro and Agol--Belolipetsky--Thomson non-arithmetic manifolds embed geodesically. Moreover, we show that the number of commensurability classes of hyperbolic manifolds with a representative of volume $\leq v$ that bounds geometrically is at least $v^{Cv}$, for $v$ large enough., Comment: 20 pages, 5 figures. Final version

We exhibit the first examples of compact orientable hyperbolic manifolds that do not have any spin structure. We show that such manifolds exist in all dimensions $n \geq 4$. The core of the argument is the construction of a compact orientable hyperbolic $4$-manifold $M$ that contains a surface $S$ of genus $3$ with self intersection $1$. The $4$-manifold $M$ has an odd intersection form and is hence not spin. It is built by carefully assembling some right angled $120$-cells along a pattern inspired by the minimum trisection of $\mathbb{C}\mathbb{P}^2$. The manifold $M$ is also the first example of a compact orientable hyperbolic $4$-manifold satisfying any of these conditions: 1) $H_2(M,\mathbb{Z})$ is not generated by geodesically immersed surfaces. 2) There is a covering $\tilde{M}$ that is a non-trivial bundle over a compact surface., Comment: 23 pages, 16 figures

We prove that there are at least 2 commensurability classes of minimal-volume hyperbolic 4-manifolds. Moreover, by applying a well-known technique due to Gromov and Piatetski-Shapiro, we build the smallest known non-arithmetic hyperbolic 4-manifold., Comment: 21 pages, 6 figures. Added the Coxeter diagrams of the commensurability classes of the manifolds. New and better proof of Lemma 2.2. Modified statements and proofs of the main theorems: now there are two commensurabilty classes of minimal volume manifolds. Typos corrected

We prove that any arithmetic hyperbolic $n$-manifold of simplest type can either be geodesically embedded into an arithmetic hyperbolic $(n+1)$-manifold or its universal $\mathrm{mod}~2$ Abelian cover can., Comment: 20 pages; revised version, typos corrected; Mathematical Research Letters vol. 25, no. 4

We show that some hyperbolic 3-manifolds which are tessellated by copies of the regular ideal hyperbolic tetrahedron embed geodesically in a complete, finite volume, hyperbolic 4-manifold. This allows us to prove that the complement of the figure-eight knot geometrically bounds a complete, finite volume hyperbolic 4-manifold. This the first example of geometrically bounding hyperbolic knot complement and, amongst known examples of geometrically bounding manifolds, the one with the smallest volume., Comment: 9 pages, 4 figures, typos corrected, improved exposition of tetrahedral manifolds. Added Proposition 3.3, which gives necessary and sufficient conditions for M_T to be a manifold, and Remark 4.4, which shows that the figure-eight knot bounds a 4-manifold of minimal volume. Updated bibliography

We develop a way of seeing a complete orientable hyperbolic $4$-manifold $\mathcal{M}$ as an orbifold cover of a Coxeter polytope $\mathcal{P} \subset \mathbb{H}^4$ that has a facet colouring. We also develop a way of finding totally geodesic sub-manifolds $\mathcal{N}$ in $\mathcal{M}$, and describing the result of mutations along $\mathcal{N}$. As an application of our method, we construct an example of a complete orientable hyperbolic $4$-manifold $\mathcal{X}$ with a single non-toric cusp and a complete orientable hyperbolic $4$-manifold $\mathcal{Y}$ with a single toric cusp. Both $\mathcal{X}$ and $\mathcal{Y}$ have twice the minimal volume among all complete orientable hyperbolic $4$-manifolds., Comment: 24 pages, 11 figures; classification in Proposition 3.2 is incomplete: see arXiv:2009.09995 for correction

In this paper, for each finite group $G$, we construct explicitly a non-compact complete finite-volume arithmetic hyperbolic $4$-manifold $M$ such that $\mathrm{Isom}\,M \cong G$, or $\mathrm{Isom}^{+}\,M \cong G$. In order to do so, we use essentially the geometry of Coxeter polytopes in the hyperbolic $4$-space, on one hand, and the combinatorics of simplicial complexes, on the other., Comment: 32 pages, 10 figures; Int. Math. Res. Notices (2015); SAGE worksheet available at https://doi.org/10.7910/DVN/0YUU6O; a minor mistake in the proof of Proposition 4.4 corrected in Proposition 2.5 / Remark 2.6 of arXiv:1710.07534

We construct here two new examples of non-orientable, non-compact, hyperbolic 4-manifolds. The first has minimal volume $v_m = 4{\pi}^2/3$ and two cusps. This example has the lowest number of cusps among known minimal volume hyperbolic 4-manifolds. The second has volume $2\cdot v_m$ and one cusp. It has lowest volume among known one-cusped hyperbolic 4-manifolds., Comment: 12 pages, 11 figures

A finite-volume hyperbolic 3-manifold geometrically bounds if it is the geodesic boundary of a finite-volume hyperbolic 4-manifold. We construct here an example of non-compact, finite-volume hyperbolic 3-manifold that geometrically bounds. The 3-manifold is the complement of a link with eight components, and its volume is roughly equal to 29.311., Comment: 23 pages, 19 figures