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Sami Douba

Publications and source records attributed to Sami Douba.

25 records · Page 2Linked to original sources
arXiv2023

Geometric and arithmetic properties of Löbell polyhedra

The Löbell polyhedra form an infinite family of compact right-angled hyperbolic polyhedra in dimension $3$. We observe, through both elementary and more conceptual means, that the ``systoles'' of the Löbell polyhedra approach $0$, so that these polyhedra give rise to particularly straightforward examples of closed hyperbolic $3$-manifolds with arbitrarily small systole, and constitute an infinite family even up to commensurability. By computing number theoretic invariants of these polyhedra, we refine the latter result, and also determine precisely which of the Löbell polyhedra are quasi-arithmetic.

math.GT↗
arXiv2021

Thin right-angled Coxeter groups in some uniform arithmetic lattices

Using a variant of an unpublished argument due to Agol, we show that an irreducible right-angled Coxeter group on ${n \geq 3}$ vertices embeds as a thin subgroup of a uniform arithmetic lattice in an indefinite orthogonal group $\mathrm{O}(p, q)$ for some $p, q \geq 1$ satisfying $p+q=n$.

math.GR↗
arXiv2021

Proper CAT(0) actions of unipotent-free linear groups

Let $Γ$ be a finitely generated group of matrices over $\mathbb{C}$. We construct an isometric action of $Γ$ on a complete CAT(0) space $X$ such that the restriction of this action to any subgroup of $Γ$ containing no nontrivial unipotent elements is well behaved. As an application, we show that if $M$ is a graph manifold that does not admit a nonpositively curved Riemannian metric, then any finite-dimensional $\mathbb{C}$-linear representation of $π_1(M)$ maps a nontrivial element of $π_1(M)$ to a unipotent matrix. In particular, the fundamental groups of such 3-manifolds do not admit any faithful finite-dimensional unitary representations.

math.GR↗
arXiv2021

Virtually unipotent curves in some non-NPC graph manifolds

Let $M$ be a graph manifold containing a single JSJ torus $T$ and whose JSJ blocks are of the form $Σ\times S^1$, where $Σ$ is a compact orientable surface with boundary. We show that if $M$ does not admit a Riemannian metric of everywhere nonpositive sectional curvature, then there is an essential curve on $T$ such that any finite-dimensional linear representation of $π_1(M)$ maps an element representing that curve to a matrix all of whose eigenvalues are roots of $1$. In particular, this shows that $π_1(M)$ does not admit a faithful finite-dimensional unitary representation, and gives a new proof that $π_1(M)$ is not linear over any field of positive characteristic.

math.GR↗
arXiv2019

2-systems of arcs on spheres with prescribed endpoints

Let $S$ be an $n$-punctured sphere, with $n \geq 3$. We prove that $\binom{n}{3}$ is the maximum size of a family of pairwise non-homotopic simple arcs on $S$ joining a fixed pair of distinct punctures of $S$ and pairwise intersecting at most twice. On the way, we show that a square annular diagram $A$ has a corner on each of its boundary paths if $A$ contains at least one square and the dual curves of $A$ are simple arcs joining the boundary paths of $A$ and pairwise intersecting at most once.

math.GT↗