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1. Words, Hausdorff dimension and randomly free groups

   https://doi.org/https://doi.org/10.1007/s00208-017-1635-y  

   Larsen, Michael; Shalev, Aner (July 2018, Mathematische Annalen)

   We bound the size of fibers of word maps in finite and residually finite groups, and derive various applications. Our main result shows that, for any word $$1 \ne w \in F_d$$ there exists $$\e > 0$$ such that if $$\Gamma$$ is a residually finite group with infinitely many non-isomorphic non-abelian upper composition factors, then all fibers of the word map $$w:\Gamma^d \rightarrow \Gamma$$ have Hausdorff dimension at most $$d -\e$$. We conclude that profinite groups $$G := \hat\Gamma$$, $$\Gamma$$ as above, satisfy no probabilistic identity, and therefore they are \emph{randomly free}, namely, for any $$d \ge 1$$, the probability that randomly chosen elements $$g_1, \ldots , g_d \in G$$ freely generate a free subgroup (isomorphic to $$F_d$$) is $$1$$. This solves an open problem from \cite{DPSS}. Additional applications and related results are also established. For example, combining our results with recent results of Bors, we conclude that a profinite group in which the set of elements of finite odd order has positive measure has an open prosolvable subgroup. This may be regarded as a probabilistic version of the Feit-Thompson theorem. 

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2. Type-I permanence

   https://doi.org/10.1090/ert/648  

   Chirvasitu, Alexandru (July 2023, Representation Theory of the American Mathematical Society)

   We prove a number of results on the survival of the type-I property under extensions of locally compact groups: (a) that given a closed normal embedding $\mathbb{N}⊴<\#comment/>\mathbb{E}$of locally compact groups and a twisted action $(\mathrm{\alpha <\#comment/>},\mathrm{\tau <\#comment/>})$thereof on a (post)liminal $C^{\ast <\#comment/>}$-algebra $A$the twisted crossed product $A{⋊<\#comment/>}_{\mathrm{\alpha <\#comment/>},\mathrm{\tau <\#comment/>}}\mathbb{E}$is again (post)liminal and (b) a number of converses to the effect that under various conditions a normal, closed, cocompact subgroup $\mathbb{N}⊴<\#comment/>\mathbb{E}$is type-I as soon as $\mathbb{E}$is. This happens for instance if $\mathbb{N}$is discrete and $\mathbb{E}$is Lie, or if $\mathbb{N}$is finitely-generated discrete (with no further restrictions except cocompactness). Examples show that there is not much scope for dropping these conditions. In the same spirit, call a locally compact group $\mathbb{G}$type-I-preserving if all semidirect products $\mathbb{N}⋊<\#comment/>\mathbb{G}$are type-I as soon as $\mathbb{N}$is, andlinearlytype-I-preserving if the same conclusion holds for semidirect products $V⋊<\#comment/>\mathbb{G}$arising from finite-dimensional $\mathbb{G}$-representations. We characterize the (linearly) type-I-preserving groups that are (1) discrete-by-compact-Lie, (2) nilpotent, or (3) solvable Lie. 

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3. THE CYCLIC GRAPH OF A Z -GROUP

   https://doi.org/10.1017/S0004972720001318  

   COSTANZO, DAVID G.; LEWIS, MARK L.; SCHMIDT, STEFANO; TSEGAYE, EYOB; UDELL, GABE (January 2020, Bulletin of the Australian Mathematical Society)

   null (Ed.)

   Abstract: For a group G, we define a graph Delta (G) by letting G^#=G\{1} be the set of vertices and by drawing an edge between distinct elements x,y in G^# if and only if the subgroup is cyclic. Recall that a Z-group is a group where every Sylow subgroup is cyclic. In this short note, we investigate Delta (G) for a Z-group G. 

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4. Efficient arithmetic regularity and removal lemmas for induced bipartite patterns.

   https://doi.org/10.19086/da  

   Alon, Noga; Fox, Jacob; Zhao, Yufei (April 2019, Discrete analysis)

   Let G be an abelian group of bounded exponent and A⊆G. We show that if the collection of translates of A has VC dimension at most d, then for every ϵ>0 there is a subgroup H of G of index at most ϵ^{−d−o(1)} such that one can add or delete at most ϵ|G| elements to/from A to make it a union of H-cosets. We also establish a removal lemma with polynomial bounds, with applications to property testing, for induced bipartite patterns in a finite abelian group with bounded exponent. 

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5. Bitcoin: A Natural Oligopoly

   Arnosti, Nick; Weinberg, S. Matthew (January 2019, Innovations in Theoretical Computer Science)

   Although Bitcoin was intended to be a decentralized digital currency, in practice, mining power is quite concentrated. This fact is a persistent source of concern for the Bitcoin community. We provide an explanation using a simple model to capture miners' incentives to invest in equipment. In our model, n miners compete for a prize of fixed size. Each miner chooses an investment q_i, incurring cost c_iq_i, and then receives reward q^{\alpha}∑_j q_j^{\alpha}, for some \alpha≥1. When c_i = c+j for all i,j, and α=1, there is a unique equilibrium where all miners invest equally. However, we prove that under seemingly mild deviations from this model, equilibrium outcomes become drastically more centralized. In particular, (a) When costs are asymmetric, if miner i chooses to invest, then miner j has market share at least 1−c_j/c_i. That is, if miner j has costs that are (e.g.) 20% lower than those of miner i, then miner j must control at least 20% of the \emph{total} mining power. (b) In the presence of economies of scale (α>1), every market participant has a market share of at least 1−1/α, implying that the market features at most α/(α−1) miners in total. We discuss the implications of our results for the future design of cryptocurrencies. In particular, our work further motivates the study of protocols that minimize "orphaned" blocks, proof-of-stake protocols, and incentive compatible protocols. 

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