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1. Torsion Invariants of complexes of groups

   https://doi.org/10.1215/00127094-2023-0024  

   Okun, Boris; Schreve, K (February 2024, Duke)

   Hain, Richard (Ed.)

   Suppose a residually finite group G acts cocompactly on a contractible complex with strict fundamental domain Q, where the stabilizers are either trivial or have normal Z-subgroups. Let dQ be the subcomplex of Q with nontrivial stabilizers. Our main result is a computation of the homology torsion growth of a chain of finite index normal subgroups of G. We show that independent of the chain, the normalized torsion limits to the torsion of dQ shifted a degree. Under milder assumptions of acyclicity of nontrivial stabilizers, we show similar formulas for the mod p-homology growth. We also obtain formulas for the universal and the usual L^2-torsion of G in terms of the torsion of stabilizers and topology of dQ. In particular, we get complete answers for right-angled Artin groups, which shows that they satisfy a torsion analogue of Lück’s approximation theorem. 

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2. Hyperbolic Groups That Are Not Commensurably Co-Hopfian

   https://doi.org/10.1093/imrn/rnaa033  

   Stark, Emily; Woodhouse, Daniel J (April 2020, International Mathematics Research Notices)

   Abstract Sela proved that every torsion-free one-ended hyperbolic group is co-Hopfian. We prove that there exist torsion-free one-ended hyperbolic groups that are not commensurably co-Hopfian. In particular, we show that the fundamental group of every simple surface amalgam is not commensurably co-Hopfian. 

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3. Counting integral points on symmetric varieties with applications to arithmetic statistics

   https://doi.org/10.1112/plms.70039  

   Shankar, Arul; Siad, Artane; Swaminathan, Ashvin A (April 2025, Proceedings of the London Mathematical Society)

   Abstract In this article, we combine Bhargava's geometry‐of‐numbers methods with the dynamical point‐counting methods of Eskin–McMullen and Benoist–Oh to develop a new technique for counting integral points on symmetric varieties lying within fundamental domains for coregular representations. As applications, we study the distribution of the 2‐torsion subgroup of the class group in thin families of cubic number fields, as well as the distribution of the 2‐Selmer groups in thin families of elliptic curves over . For example, our results suggest that the existence of a generator of the ring of integers with small norm has an increasing effect on the average size of the 2‐torsion subgroup of the class group, relative to the Cohen–Lenstra predictions. 

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4. Local duality for representations of finite group schemes

   https://doi.org/10.1112/s0010437x19007061  

   Benson, Dave; Iyengar, Srikanth B.; Krause, Henning; Pevtsova, Julia (February 2019, Compositio Mathematica)

   A duality theorem for the stable module category of representations of a finite group scheme is proved. One of its consequences is an analogue of Serre duality, and the existence of Auslander–Reiten triangles for the $$\mathfrak{p}$$ -local and $$\mathfrak{p}$$ -torsion subcategories of the stable category, for each homogeneous prime ideal $$\mathfrak{p}$$ in the cohomology ring of the group scheme. 

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5. 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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