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Margulis Lemma, entropy and free products

Filippo Cerocchi (2014)

Annales de l’institut Fourier

We prove a Margulis’ Lemma à la Besson-Courtois-Gallot, for manifolds whose fundamental group is a nontrivial free product A * B , without 2-torsion. Moreover, if A * B is torsion-free we give a lower bound for the homotopy systole in terms of upper bounds on the diameter and the volume-entropy. We also provide examples and counterexamples showing the optimality of our assumption. Finally we give two applications of this result: a finiteness theorem and a volume estimate for reducible manifolds.

Matrix coefficients, counting and primes for orbits of geometrically finite groups

Amir Mohammadi, Hee Oh (2015)

Journal of the European Mathematical Society

Let G : = SO ( n , 1 ) ∘ and Γ ( n - 1 ) / 2 for n = 2 , 3 and when δ > n - 2 for n ≥ 4 , we obtain an effective archimedean counting result for a discrete orbit of Γ in a homogeneous space H ∖ G where H is the trivial group, a symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family { ℬ T ⊂ H ∖ G } of compact subsets, there exists η > 0 such that # [ e ] Γ ∩ ℬ T = ℳ ( ℬ T ) + O ( ℳ ( ℬ T ) 1 - η ) for an explicit measure ℳ on H ∖ G which depends on Γ . We also apply the affine sieve and describe the distribution of almost primes on orbits of Γ in arithmetic settings....

Matrix identities involving multiplication and transposition

Karl Auinger, Igor Dolinka, Michael V. Volkov (2012)

Journal of the European Mathematical Society

We study matrix identities involving multiplication and unary operations such as transposition or Moore–Penrose inversion. We prove that in many cases such identities admit no finite basis.

Maximal clones and maximal permutation groups

Péter P. Pálfy (2007)

Discussiones Mathematicae - General Algebra and Applications

A fundamental result in universal algebra is the theorem of Rosenberg describing the maximal subclones in the clone of all operations over a finite set. In group theory, the maximal subgroups of the symmetric groups are classified by the O'Nan-Scott Theorem. We shall explore the similarities and differences between these two analogous major results. In addition, we show that a primitive permutation group of diagonal type can be maximal in the symmetric group only if its socle is the direct product...

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