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1. Dimension estimates for the set of points with non-dense orbit in homogeneous spaces

   https://doi.org/10.1007/s00209-019-02386-7  

   Kleinbock, D. (January 2020, Mathematische Zeitschrift)

   Let 𝑋=𝐺/Γ, where G is a Lie group and Γ is a lattice in G, and let U be a subset of X whose complement is compact. We use the exponential mixing results for diagonalizable flows on X to give upper estimates for the Hausdorff dimension of the set of points whose trajectories miss U. This extends a recent result of Kadyrov et al. (Dyn Syst 30(2):149–157, 2015) and produces new applications to Diophantine approximation, such as an upper bound for the Hausdorff dimension of the set of weighted uniformly badly approximable systems of linear forms, generalizing an estimate due to Broderick and Kleinbock (Int J Number Theory 11(7):2037–2054, 2015). 

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2. A geometric formula for multiplicities of 𝐾-types of tempered representations

   https://doi.org/10.1090/tran/7857  

   Hochs, Peter; Song, Yanli; Yu, Shilin (December 2019, Transactions of the American Mathematical Society)

   Let G G be a connected, linear, real reductive Lie group with compact centre. Let K > G K>G be compact. Under a condition on K K , which holds in particular if K K is maximal compact, we give a geometric expression for the multiplicities of the K K -types of any tempered representation (in fact, any standard representation) π \pi of G G . This expression is in the spirit of Kirillov’s orbit method and the quantisation commutes with reduction principle. It is based on the geometric realisation of π | K \pi |_K obtained in an earlier paper. This expression was obtained for the discrete series by Paradan, and for tempered representations with regular parameters by Duflo and Vergne. We obtain consequences for the support of the multiplicity function, and a criterion for multiplicity-free restrictions that applies to general admissible representations. As examples, we show that admissible representations of SU ( p , 1 ) \textrm {SU}(p,1) , SO 0 ( p , 1 ) \textrm {SO}_0(p,1) , and SO 0 ( 2 , 2 ) \textrm {SO}_0(2,2) restrict multiplicity freely to maximal compact subgroups. 

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3. Dimension drop for diagonalizable flows on homogeneous spaces

   https://doi.org/10.3934/jmd.2024012  

   Kleinbock, Dmitry; Mirzadeh, Shahriar (January 2024, Journal of Modern Dynamics)

   Let X =G/Γ, where G is a connected Lie group and Γ is a lattice in G. Let O be an open subset of X, and let F = {g_t : t ≥ 0} be a one-parameter subsemigroup of G. Consider the set of points in X whose F-orbit misses O; it has measure zero if the flow is ergodic. It has been conjectured that, assuming ergodicity, this set has Hausdorff dimension strictly smaller than the dimension of X. This conjecture has been proved when X is compact or when G is a simple Lie group of real rank 1, or, most recently, for certain flows on the space of lattices. In this paper we prove this conjecture for arbitrary Addiagonalizable flows on irreducible quotients of semisimple Lie groups. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on G/Γ. We also derive an application to jointly Dirichlet-Improvable systems of linear forms. 

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4. Classifying primitive solvable permutation groups of rank 5 and 6

   https://doi.org/10.1515/jgth-2023-0205  

   Dey, Anakin; O’Neal, Kolton; Tran, Duc_Van Khanh; Upshur, Camron; Yang, Yong (April 2024, Journal of Group Theory)

   Abstract Let 𝐺 be a finite solvable permutation group acting faithfully and primitively on a finite set Ω.Let $G_0$G_{0}be the stabilizer of a point 𝛼 in Ω.The rank of 𝐺 is defined as the number of orbits of $G_0$G_{0}in Ω, including the trivial orbit $\left\{\alpha \right\}$\{\alpha\}.In this paper, we completely classify the cases where 𝐺 has rank 5 and 6, continuing the previous works on classifying groups of rank 4 or lower. 

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5. 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)

   Abstract 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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