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1. Grothendieck--Witt groups of some singular schemes

   Karoubi, M.; Schlichting, M.; Weibel, C. (July 2021, Proceedings of the London Mathematical Society)

   null (Ed.)

   We establish some structural results for the Witt and Grothendieck--Witt groups of schemes over \Z[1/2], including homotopy invariance for Witt groups and a formula for the Witt and Grothendieck--Witt groups of punctured affine spaces over a scheme. All these results hold for singular schemes and at the level of spectra. 

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2. On p-Group Isomorphism: Search-To-Decision, Counting-To-Decision, and Nilpotency Class Reductions via Tensors

   https://doi.org/10.4230/LIPIcs.CCC.2021.16  

   Grochow, Joshua A.; Qiao, Youming (August 2021, 36th Computational Complexity Conference (CCC 2021))

   Kabanets, Valentine (Ed.)

   In this paper we study some classical complexity-theoretic questions regarding Group Isomorphism (GpI). We focus on p-groups (groups of prime power order) with odd p, which are believed to be a bottleneck case for GpI, and work in the model of matrix groups over finite fields. Our main results are as follows. - Although search-to-decision and counting-to-decision reductions have been known for over four decades for Graph Isomorphism (GI), they had remained open for GpI, explicitly asked by Arvind & Torán (Bull. EATCS, 2005). Extending methods from Tensor Isomorphism (Grochow & Qiao, ITCS 2021), we show moderately exponential-time such reductions within p-groups of class 2 and exponent p. - Despite the widely held belief that p-groups of class 2 and exponent p are the hardest cases of GpI, there was no reduction to these groups from any larger class of groups. Again using methods from Tensor Isomorphism (ibid.), we show the first such reduction, namely from isomorphism testing of p-groups of "small" class and exponent p to those of class two and exponent p. For the first results, our main innovation is to develop linear-algebraic analogues of classical graph coloring gadgets, a key technique in studying the structural complexity of GI. Unlike the graph coloring gadgets, which support restricting to various subgroups of the symmetric group, the problems we study require restricting to various subgroups of the general linear group, which entails significantly different and more complicated gadgets. The analysis of one of our gadgets relies on a classical result from group theory regarding random generation of classical groups (Kantor & Lubotzky, Geom. Dedicata, 1990). For the nilpotency class reduction, we combine a runtime analysis of the Lazard Correspondence with Tensor Isomorphism-completeness results (Grochow & Qiao, ibid.). 

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3. Some groups with planar boundaries, in ``Surveys in 3-manifold topology and geometry".

   Kim, Sang-hyun; Walsh, Genevieve (December 2022, Surv. Differ. Geom.)

   In this expository note, we illustrate phenomena and conjectures about boundaries of hyperbolic groups by considering the special cases of certain amalgams of hyperbolic groups. While doing so, we describe fundamental results on hyperbolic groups and their boundaries by Bowditch [5] and Haissinsky [27], along with special treatments for the boundaries of free groups by Otal [48] and Cashen [15]. 

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4. Patterson–Sullivan measures for transverse subgroups

   https://doi.org/10.3934/jmd.2024009  

   Canary, Richard; Zhang, Tengren; Zimmer, Andrew (January 2024, Journal of Modern Dynamics)

   Forni, Giovanni (Ed.)

   We study Patterson–Sullivan measures for a class of discrete subgroups of higher rank semisimple Lie groups, called transverse groups, whose limit set is well-defined and transverse in a partial flag variety. This class of groups includes both Anosov and relatively Anosov groups, as well as all discrete subgroups of rank one Lie groups. We prove an analogue of the Hopf–Tsuji–Sullivan dichotomy and then use this dichotomy to prove a variant of Burger's Manhattan Curve Theorem. We also use the Patterson–Sullivan measures to obtain conditions for when a subgroup has critical exponent strictly less than the original transverse group. These gap results are new even for Anosov groups. 

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5. Compactifications of pseudofinite and pseudoamenable groups

   https://doi.org/10.4171/GGD/852  

   Conant, Gabriel; Hrushovski, Ehud; Pillay, Anand (January 2025, Groups, Geometry, and Dynamics)

   We first give simplified and corrected accounts of some results in work by Pillay (2017) on compactifications of pseudofinite groups. For instance, we use a classical theorem of Turing (1938) to give a simplified proof that any definable compactification of a pseudofinite group has an abelian connected component. We then discuss the relationship between Turing’s work, the Jordan–Schur theorem, and a (relatively) more recent result of Kazhdan (1982) on approximate homomorphisms, and we use this to widen our scope from finite groups to amenable groups. In particular, we develop a suitable continuous logic framework for dealing with definable homomorphisms from pseudoamenable groups to compact Lie groups. Together with the stabilizer theorems of Hrushovski (2012) and Montenegro et al. (2020), we obtain a uniform (but non-quantitative) analogue of Bogolyubov’s lemma for sets of positive measure in discrete amenable groups. We conclude with brief remarks on the case of amenable topological groups. 

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