Markoff triples and strong approximation

Brauer-Manin obstruction for Markoff surfaces

“…for p = 3 and ord 3 (d) = 1 ( 1 2 Z/Z) 3 for p = 3 and ord 3 (d) ≥ 3 ( 1 2 Z/Z) 3 \ (0, 0, 0) for p = 5 and ord 5 (d) = 1 ( 1 2 Z/Z) 3 for p = 5 and ord 5 (d) ≥ 3.…”

Cycle integrals of modular functions, Markov geodesics and a conjecture of Kaneko

“…Since no zigzag paths are allowed, each branch has a unique orientation. For example, the branch with the quadratics (3, 2, 3, 4), (3, 2, 3 2 , 4), (3, 2, 3 3 , 4) is a left branch, whereas the branch with (3, 2, 3, 4), (3,2,4,2,3,4), (3, (2, 4) 2 , 2, 3, 4) is a right branch. We call the first vertex at the top of any branch its tip.…”

“…We call the first vertex at the top of any branch its tip. Except for the two singular cases of (2, 3) and (3,2,4), each Markov number lies both on a right and a left branch but it is the tip of only a left or a right branch, except for (3,2,3,4) which is the tip of both the leftmost and the rightmost branches.…”

“…(See for example Corollary 1.27 in [1].) Markov numbers arise in many different contexts: see [3], [4], [5] for some recent developments regarding the Markov surfaces.…”

Conjugacy growth of commutators

“…In certain contexts, it is more natural to consider the growth rate of the conjugacy classes of G. For a given conjugacy class C of G, define the length of C by inf g∈C length(g), Date: February 12, 2019. 1 and define ∂B conj k (G, S) as the set of conjugacy classes of G with length k. In the case of F r , the minimal-length elements of a conjugacy class are precisely its cyclically reduced elements, all of which are cyclic conjugates of each other. The conjugacy growth of F r can be described as ∂B conj k (G, S) ∼ (2r − 1) k /k, which agrees with the intuition of identifying the cyclic conjugates among the 2r(2r − 1) k−1 words of length k; for the full explicit formula, see [19,Proposition 17.8].…”