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1. Elimination of imaginaries in C((Γ))$\mathbb {C}((\Gamma ))$

   https://doi.org/10.1112/jlms.12751  

   Vicaría, Mariana ( May 2023 , Journal of the London Mathematical Society)

   In this paper, we study elimination of imaginaries in henselian valued fields of equicharacteristic zero and residue field algebraically closed. The results are sensitive to the complexity of the value group. We focus first on the case where the ordered abelian group has finite spines, and then prove a better result for the dp‐minimal case. In previous work the author proved that an ordered abelian with finite spines weakly eliminates imaginaries once one adds sorts for the quotient groups for each definable convex subgroup , and sorts for the quotient groups where is a definable convex subgroup and . We refer to these sorts as thequotient sorts. Jahnke, Simon, and Walsberg (J. Symb. Log.82(2017) 151–165) characterized ‐minimal ordered abelian groups as those without singular primes, that is, for every prime number one has .

   We prove the following two theorems:

    

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2. Representations and Tensor Product Growth

   https://doi.org/10.1093/imrn/rnac172  

   Larsen, Michael ; Shalev, Aner ; Tiep, Pham Huu ( July 2022 , International Mathematics Research Notices)

   The deep theory of approximate subgroups establishes three-step product growth for subsets of finite simple groups $G$ of Lie type of bounded rank. In this paper, we obtain two-step growth results for representations of such groups $G$ (including those of unbounded rank), where products of subsets are replaced by tensor products of representations. Let $G$ be a finite simple group of Lie type and $\chi $ a character of $G$. Let $|\chi |$ denote the sum of the squares of the degrees of all (distinct) irreducible characters of $G$ that are constituents of $\chi $. We show that for all $\delta>0$, there exists $\epsilon>0$, independent of $G$, such that if $\chi $ is an irreducible character of $G$ satisfying $|\chi | \le |G|^{1-\delta }$, then $|\chi ^2| \ge |\chi |^{1+\epsilon }$. We also obtain results for reducible characters and establish faster growth in the case where $|\chi | \le |G|^{\delta }$. In another direction, we explore covering phenomena, namely situations where every irreducible character of $G$ occurs as a constituent of certain products of characters. For example, we prove that if $|\chi _1| \cdots |\chi _m|$ is a high enough power of $|G|$, then every irreducible character of $G$ appears in $\chi _1\cdots \chi _m$. Finally, we obtain growth results for compact semisimple Lie groups.

    

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3. Characters of $$\pi '$$-degree and small cyclotomic fields

   https://doi.org/10.1007/s10231-020-01025-x  

   Giannelli, Eugenio ; Hung, Nguyen Ngoc ; Schaeffer Fry, A. A. ; Vallejo, Carolina ( June 2021 , Annali di Matematica Pura ed Applicata (1923 -))

   null (Ed.)

   Abstract We show that every finite group of order divisible by 2 or q , where q is a prime number, admits a $$\{2, q\}'$$ { 2 , q } ′ -degree nontrivial irreducible character with values in $${\mathbb{Q}}(e^{2 \pi i /q})$$ Q ( e 2 π i / q ) . We further characterize when such character can be chosen with only rational values in solvable groups. These results follow from more general considerations on groups admitting a $$\{p, q\}'$$ { p , q } ′ -degree nontrivial irreducible character with values in $${\mathbb{Q}}(e^{2 \pi i /p})$$ Q ( e 2 π i / p ) or $${\mathbb{Q}}(e^{ 2 \pi i/q})$$ Q ( e 2 π i / q ) , for any pair of primes p and q . Along the way, we completely describe simple alternating groups admitting a $$\{p, q\}'$$ { p , q } ′ -degree nontrivial irreducible character with rational values. 

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4. Coarse models of homogeneous spaces and translations-like actions

   D. B. McReynolds, Mark Pengitore ( January 2019 , Preprint)

   For finitely generated groups G and H equipped with word metrics, a translation-like action of H on G is a free action where each element of H moves elements of G a bounded distance. Translation-like actions provide a geometric generalization of subgroup containment. Extending work of Cohen, we show that cocompact lattices in a general semisimple Lie group G that is not isogenous to SL(2,ℝ) admit translation-like actions by ℤ2. This result follows from a more general result. Namely, we prove that any cocompact lattice in the unipotent radical N of the Borel subgroup AN of G acts translation-like on any cocompact lattice in G. We also prove that for noncompact simple Lie groups G,H with H more » « less
5. Hyperbolic families and coloring graphs on surfaces

   https://doi.org/https://doi.org/10.1090/btran/26  

   Postle, Luke ; Thomas, Robin ( December 2018 , Transactions of the American Mathematical Society. Series B)

   We develop a theory of linear isoperimetric inequalities for graphs on surfaces and apply it to coloring problems, as follows. Let $ G$ be a graph embedded in a fixed surface $ \Sigma $ of genus $ g$ and let $ L=(L(v):v\in V(G))$ be a collection of lists such that either each list has size at least five, or each list has size at least four and $ G$ is triangle-free, or each list has size at least three and $ G$ has no cycle of length four or less. An $ L$-coloring of $ G$ is a mapping $ \phi $ with domain $ V(G)$ such that $ \phi (v)\in L(v)$ for every $ v\in V(G)$ and $ \phi (v)\ne \phi (u)$ for every pair of adjacent vertices $ u,v\in V(G)$. We prove if every non-null-homotopic cycle in $ G$ has length $ \Omega (\log g)$, then $ G$ has an $ L$-coloring, if $ G$ does not have an $ L$-coloring, but every proper subgraph does (``$ L$-critical graph''), then $ \vert V(G)\vert=O(g)$, if every non-null-homotopic cycle in $ G$ has length $ \Omega (g)$, and a set $ X\subseteq V(G)$ of vertices that are pairwise at distance $ \Omega (1)$ is precolored from the corresponding lists, then the precoloring extends to an $ L$-coloring of $ G$, if every non-null-homotopic cycle in $ G$ has length $ \Omega (g)$, and the graph $ G$ is allowed to have crossings, but every two crossings are at distance $ \Omega (1)$, then $ G$ has an $ L$-coloring, if $ G$ has at least one $ L$-coloring, then it has at least $ 2^{\Omega (\vert V(G)\vert)}$ distinct $ L$-colorings. We show that the above assertions are consequences of certain isoperimetric inequalities satisfied by $ L$-critical graphs, and we study the structure of families of embedded graphs that satisfy those inequalities. It follows that the above assertions hold for other coloring problems, as long as the corresponding critical graphs satisfy the same inequalities. 

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