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1. On the atomic structure of torsion-free monoids

   https://doi.org/10.1007/s00233-023-10385-8  

   Gotti, Felix; Vulakh, Joseph (October 2023, Semigroup Forum)

   Abstract LetMbe a cancellative and commutative (additive) monoid. The monoidMis atomic if every non-invertible element can be written as a sum of irreducible elements, which are also called atoms. Also,Msatisfies the ascending chain condition on principal ideals (ACCP) if every increasing sequence of principal ideals (under inclusion) becomes constant from one point on. In the first part of this paper, we characterize torsion-free monoids that satisfy the ACCP as those torsion-free monoids whose submonoids are all atomic. A submonoid of the nonnegative cone of a totally ordered abelian group is often called a positive monoid. Every positive monoid is clearly torsion-free. In the second part of this paper, we study the atomic structure of certain classes of positive monoids. 

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2. Convex co‐compact representations of 3‐manifold groups

   https://doi.org/10.1112/topo.12332  

   Islam, Mitul; Zimmer, Andrew (May 2024, Journal of Topology)

   Abstract A representation of a finitely generated group into the projective general linear group is called convex co‐compact if it has finite kernel and its image acts convex co‐compactly on a properly convex domain in real projective space. We prove that the fundamental group of a closed irreducible orientable 3‐manifold can admit such a representation only when the manifold is geometric (with Euclidean, Hyperbolic or Euclidean Hyperbolic geometry) or when every component in the geometric decomposition is hyperbolic. In each case, we describe the structure of such examples. 

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3. Characterizing candidates for Cannon's conjecture from geometric measure theory

   https://doi.org/10.1112/blms.12814  

   Cheetham‐West, Tamunonye; Nolte, Alexander (February 2023, Bulletin of the London Mathematical Society)

   Abstract We show that recent work of Song implies that torsion‐free hyperbolic groups with Gromov boundary arerealized as fundamental groups of closed 3‐manifolds of constant negative curvature if and only if the solution to an associated spherical Plateau problem for group homology is isometric to such a 3‐manifold, and suggest some related questions. 

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4. Bounded projections to the Z$$\mathcal {Z}$$‐factor graph

   https://doi.org/10.1112/topo.70024  

   Clay, Matt; Uyanik, Caglar (June 2025, Journal of Topology)

   Abstract Suppose that is a free product , where each of the groups is torsion‐free and is a free group of rank . Let be the deformation space associated to this free product decomposition. We show that the diameter of the projection of the subset of where a given element has bounded length to the ‐factor graph is bounded, where the diameter bound depends only on the length bound. This relies on an analysis of the boundary of as a hyperbolic group relative to the collection of subgroups together with a given nonperipheral cyclic subgroup. The main theorem is new even in the case that , in which case is the Culler–Vogtmann outer space. In a future paper, we will apply this theorem to study the geometry of free group extensions. 

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5. Shapes of hyperbolic triangles and once-punctured torus groups

   https://doi.org/10.1007/s00209-021-02745-3  

   Kim, Sang-hyun; Koberda, Thomas; Lee, Jaejeong; Ohshika, Ken’ichi; Tan, Ser Peow; Gao, Xinghua (January 2021, Mathematische Zeitschrift)

   null (Ed.)

   Abstract Let $$\Delta $$ Δ be a hyperbolic triangle with a fixed area $$\varphi $$ φ . We prove that for all but countably many $$\varphi $$ φ , generic choices of $$\Delta $$ Δ have the property that the group generated by the $$\pi $$ π -rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all $$\varphi \in (0,\pi ){\setminus }\mathbb {Q}\pi $$ φ ∈ ( 0 , π ) \ Q π , a dense set of triangles does afford nontrivial relations, which in the generic case map to hyperbolic translations. To establish this fact, we study the deformation space $$\mathfrak {C}_\theta $$ C θ of singular hyperbolic metrics on a torus with a single cone point of angle $$\theta =2(\pi -\varphi )$$ θ = 2 ( π - φ ) , and answer an analogous question for the holonomy map $$\rho _\xi $$ ρ ξ of such a hyperbolic structure $$\xi $$ ξ . In an appendix by Gao, concrete examples of $$\theta $$ θ and $$\xi \in \mathfrak {C}_\theta $$ ξ ∈ C θ are given where the image of each $$\rho _\xi $$ ρ ξ is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3-manifolds. 

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