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1. The spread of a finite group

   https://doi.org/doi.org/10.4007/annals.2021.193.2.5  

   Burness, Tim; Guralnick, Robert; Harper, Scott (January 2021, Annals of mathematics)

   null (Ed.)

   A group G is said to be 3/2-generated if every nontrivial element belongs to a generating pair. It is easy to see that if G has this property, then every proper quotient of G is cyclic. In this paper we prove that the converse is true for finite groups, which settles a conjecture of Breuer, Guralnick and Kantor from 2008. In fact, we prove a much stronger result, which solves a problem posed by Brenner and Wiegold in 1975. Namely, if G is a finite group and every proper quotient of G is cyclic, then for any pair of nontrivial elements x1, x2 ϵ G, there exists y ϵ G such that G = ⟨x1, y⟩ = ⟨x2, y⟩. In other words, s(G) ⩾ 2, where s(G) is the spread of G. Moreover, if u(G) denotes the more restrictive uniform spread of G, then we can completely characterise the finite groups G with u(G) = 0 and u(G) = 1. To prove these results, we first establish a reduction to almost simple groups. For simple groups, the result was proved by Guralnick and Kantor in 2000 using probabilistic methods, and since then the almost simple groups have been the subject of several papers. By combining our reduction theorem and this earlier work, it remains to handle the groups with socle an exceptional group of Lie type, and this is the case we treat in this paper. 

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2. Equilibrium and nonequilibrium Gibbs states on sofic groups

   https://doi.org/10.1088/1361-6544/adb236  

   Shriver, Christopher (February 2025, Nonlinearity)

   Abstract Recent work of Barbieri and Meyerovitch has shown that, for very general spin systems indexed by sofic groups, equilibrium (i.e. pressure-maximizing) states are Gibbs. The main goal of this paper is to show that the converse fails in an interesting way: for the Ising model on a free group, the free-boundary state typically fails to be equilibrium as long as it is not theonlyGibbs state. For every temperature between the uniqueness and reconstruction thresholds a typical sofic approximation gives this state finite but non-maximal pressure, and for every lower temperature the pressure is non-maximal overeverysofic approximation. We also show that, for more general interactions on sofic groups, the local on average limit of Gibbs states over a sofic approximation Σ, if it exists, is a mixture of Σ-equilibrium states. We use this to show that the plus- and minus-boundary-condition Ising states are Σ-equilibrium if Σ is any sofic approximation to a free group. Combined with a result of Dembo and Montanari, this implies that these states have the same entropy over every sofic approximation. 

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3. A weaker notion of the finite factorization property

   Jiang, Henry; Kanungo, Shihan; Kim, Harry (April 2024, Communications of the Korean Mathematical Society)

   An (additive) commutative monoid is called atomic if every given non-invertible element can be written as a sum of atoms (i.e., irreducible elements), in which case, such a sum is called a factorization of the given element. The number of atoms (counting repetitions) in the corresponding sum is called the length of the factorization. Following Geroldinger and Zhong, we say that an atomic monoid M is a length-finite factorization monoid if each b ∈ M has only finitely many factorizations of any prescribed length. An additive submonoid of ℝ≥0 is called a positive monoid. Factorizations in positive monoids have been actively studied in recent years. The main purpose of this paper is to give a better understanding of the non-unique factorization phenomenon in positive monoids through the lens of the length-finite factorization property. To do so, we identify a large class of positive monoids which satisfy the length-finite factorization property. Then we compare the length-finite factorization property to the bounded and the finite factorization properties, which are two properties that have been systematically investigated for more than thirty years. 

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4. Counting, equidistribution and entropy gaps at infinity with applications to cusped Hitchin representations

   https://doi.org/10.1515/crelle-2022-0035  

   Bray, Harrison; Canary, Richard; Kao, Lien-Yung; Martone, Giuseppe (August 2022, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We show that if an eventually positive, non-arithmetic, locally Hölder continuous potential for a topologically mixingcountable Markov shift with (BIP) has an entropy gap at infinity,then one may apply the renewal theorem of Kesseböhmer and Kombrink to obtain counting and equidistributionresults. We apply these general results to obtain counting and equidistribution results for cusped Hitchinrepresentations, and more generally for cusped Anosov representations of geometrically finite Fuchsian groups. 

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5. Strongly Compact Cardinals and Ordinal Definability

   https://doi.org/10.1142/S0219061322500106  

   Goldberg, Gabriel (May 2023, Journal of Mathematical Logic)

   This paper explores several topics related to Woodin’s HOD conjecture. We improve the large cardinal hypothesis of Woodin’s HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a strongly compact cardinal and the HOD hypothesis holds, there is no elementary embedding from HOD to HOD, settling a question of Woodin. We show that the HOD hypothesis is equivalent to a uniqueness property of elementary embeddings of levels of the cumulative hierarchy. We prove that the HOD hypothesis holds if and only if every regular cardinal above the first strongly compact cardinal carries an ordinal definable omega Jónsson algebra. We show that if the HOD hypothesis holds and HOD satisfies the Ultrapower Axiom, then every supercompact cardinal is supercompact in HOD. 

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