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Let G be a linear algebraic group acting linearly on a vector space (or more generally, an affine variety) $ \mathcal {V} $ V , and let $ {\Bbbk [\mathcal {V}]^{G}} $ k [ V ] G be the corresponding algebra of invariant polynomial functions. A separating set $ S \subseteq {\Bbbk [\mathcal {V}]^{G}} $ S ⊆ k [ V ] G is a set of polynomials with the property that for all $ v,w \in \mathcal {V} $ v , w ∈ V , if there exists $ f \in {\Bbbk [\mathcal {V}]^{G}} $ f ∈ k [ V ] G separating $v$ and $w$, then there exists $ f \in S $ f ∈ S separating $v$ and $w$. In this article, we consider the action of $ G = \operatorname {GL}_2(\mathbb {C}) $ G = GL 2 ⁡ (C) on the $ \mathbb {C} $ C -vector space $ {\mathcal {M}}_2^n $ M 2 n of n-tuples of $ 2 \times 2 $ 2 × 2 matrices by simultaneous conjugation. Minimal generating sets $ S_n $ S n of $ \mathbb {C}[{\mathcal {M}}_2^n]^G $ C [ M 2 n ] G are well known and $ |S_n| = \frac 16(n^3+11n) $ | S n | = 1 6 (n 3 + 11 n). In recent work, Kaygorodov et al. [Kaygorodov I, Lopatin A, Popov Y. Separating invariants for $ 2 \times 2 $ 2 × 2 matrices. Linear Algebra Appl. 2018;559:114-124.] showed that for all $ n \geq ~1 $ n ≥ 1 , $ S_n $ S n is a minimal separating set by inclusion, i.e. that no proper subset of $ S_n $ S n is a separating set. This does not necessarily mean that $ S_n $ S n has minimum cardinality among all separating sets for $ \mathbb {C}[{\mathcal {M}}_2^n]^G $ C [ M 2 n ] G . Our main result shows that any separating set for $ \mathbb {C}[{\mathcal {M}}_2^n]^G $ C [ M 2 n ] G has cardinality $ \geq ~5n-5 $ ≥ 5 n − 5. In particular, there is no separating set of size $ \dim (\mathbb {C}[{\mathcal {M}}_2^n]^G) = 4n-3 $ dim ⁡ (C [ M 2 n ] G) = 4 n − 3 for $ n \geq ~3 $ n ≥ 3. Further, $ S_3 $ S 3 has indeed minimum cardinality as a separating set, but for $ n \geq ~4 $ n ≥ 4 there may exist a smaller separating set than $ S_n $ S n . We show that a smaller separating set does in fact exist for all $ n \geq ~5 $ n ≥ 5. We also prove similar results for the left–right action of $ \operatorname {SL}_2(\mathbb {C}) \times \operatorname {SL}_2(\mathbb {C}) $ SL 2 ⁡ (C) × SL 2 ⁡ (C) on $ {\mathcal {M}}_2^n $ M 2 n . [ABSTRACT FROM AUTHOR]

We prove effective equidistribution of horospherical flows in $\operatorname {SO}(n,1)^{\circ } / \Gamma $ when $\Gamma $ is geometrically finite and the frame flow is exponentially mixing for the Bowen–Margulis–Sullivan measure. We also discuss settings in which such an exponential mixing result is known to hold. As part of the proof, we show that the Patterson–Sullivan measure satisfies some friendly like properties when $\Gamma $ is geometrically finite. [ABSTRACT FROM AUTHOR]

We classify anticommutative algebras in a specific closed subvariety and describe all degenerations between these algebras. Anticommutative n -dimensional algebras of the third degeneration level are classified for any natural n using these results. [ABSTRACT FROM AUTHOR]

n-ary algebras of the first degeneration level are described. A detailed classification is given in the cases n = 2 , 3 . [ABSTRACT FROM AUTHOR]

Anticommutative Engel algebras of the first five degeneration levels are classified. All algebras appearing in this classification are nilpotent Malcev algebras. [ABSTRACT FROM AUTHOR]