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Abstract Let M be an irreducible $$3$$ -manifold M with empty or toroidal boundary which has at least one hyperbolic piece in its geometric decomposition, and let A be a finite abelian group. Generalizing work of Sun [20] and of Friedl–Herrmann [7], we prove that there exists a finite cover $$M' \to M$$ so that A is a direct factor in $$H_1(M',{\mathbb Z})$$ .more » « less
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null (Ed.)Abstract The purpose of the present paper is to prove existence of super-exponentially many compact orientable hyperbolic arithmetic $$n$$-manifolds that are geometric boundaries of compact orientable hyperbolic $(n+1)$-manifolds, for any $$n \geq 2$$, thereby establishing that these classes of manifolds have the same growth rate with respect to volume as all compact orientable hyperbolic arithmetic $$n$$-manifolds. An analogous result holds for non-compact orientable hyperbolic arithmetic $$n$$-manifolds of finite volume that are geometric boundaries for $$n \geq 2$$.more » « less
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Collin, Olivier; Friedl, Stefan; Gordon, Cameron; Tillmann, Stephan; Watson, Liam (Ed.)The automorphism groups of integral Lorentzian lattices act by isometries on hyperbolic space with finite covolume. In the case of reflective integral lattices, the automorphism groups are commensurable to arithmetic hyperbolic reflection groups. However, for a fixed dimension, there is only finitely many reflective integral Lorentzian lattices, and these can only occur in small dimensions. The goal of this note is to construct embeddings of lowdimensional integral Lorentzian lattices into unimodular Lorentzian lattices associated to right-angled reflection groups. As an application, we construct many discrete groups of Isom(Hn) for small n which are C-special in the sense of Haglund-Wise.more » « less
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Fukui, Toshizumi; Koike, Satoshi; Paunescu, Laurențiu (Ed.)The Heegaard genus of a 3-manifold, as well as the growth of Heegaard genus in its finite sheeted cover spaces, has extensively been studied in terms of algebraic, geometric and topological properties of the 3-manifold. This note shows that analogous results concerning the trisection genus of a smooth, orientable 4-manifold have more general answers than their counterparts for 3-manifolds. In the case of hyperbolic 4-manifolds, upper and lower bounds are given in terms of volume and a trisection of the Davis manifold is described.more » « less
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