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Counting arithmetic subgroups and subgroup growth of virtually free groups

Amichai Eisenmann (2015)

Journal of the European Mathematical Society

Let K be a p -adic field, and let H = P S L 2 ( K ) endowed with the Haar measure determined by giving a maximal compact subgroup measure 1 . Let A L H ( x ) denote the number of conjugacy classes of arithmetic lattices in H with co-volume bounded by x . We show that under the assumption that K does not contain the element ζ + ζ - 1 , where ζ denotes the p -th root of unity over p , we have lim x log A L H ( x ) x log x = q - 1 where q denotes the order of the residue field of K .

Covering locally compact groups by less than 2 ω many translates of a compact nullset

Márton Elekes, Árpád Tóth (2007)

Fundamenta Mathematicae

Gruenhage asked if it was possible to cover the real line by less than continuum many translates of a compact nullset. Under the Continuum Hypothesis the answer is obviously negative. Elekes and Stepr mans gave an affirmative answer by showing that if C E K is the well known compact nullset considered first by Erdős and Kakutani then ℝ can be covered by cof() many translates of C E K . As this set has no analogue in more general groups, it was asked by Elekes and Stepr mans whether such a result holds for...

Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane*

Yuri L. Sachkov (2011)

ESAIM: Control, Optimisation and Calculus of Variations

The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is considered. In the previous works [Moiseev and Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009004; Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009031], extremal trajectories were defined, their local and global optimality were studied. In this paper the global structure of the exponential mapping is described. On this basis an explicit characterization of the cut locus and Maxwell set is obtained....

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