Fine asymptotic geometry in the Heisenberg group

Isometric embeddings into Heisenberg groups

“…By left-invariance it is enough to show that all infinite geodesics in (H 1 , d SF ) which pass through the origin are straight lines. By [3, Theorem 1], [28, §4] it is known that if · is strictly convex, then the geodesics in (H 1 , d SF ) passing through 0 project to the (x, y)-plane either to (i) straight lines or line segments, or (ii) isoperimetric paths passing through zero, see also Section 2.3 in [12]. By an isoperimetric path we mean a subpath of a dilated and left-translated isoperimetrix in the sense of Buseman.…”

Statistical hyperbolicity of relatively hyperbolic groups

“…We consider the annular-like set A tn,n := ∪ tn≤i≤n S i (G) × S n−i (H). By Lemma 4.1, we know that (8) |A tn,n |/|S n | → 1 as n → ∞.…”

“…Recall that A tn,n = ∪ tn≤j≤n S j (G)×S n−j (H). Note that C R ⊂ S n and by (8), |S n |/|A tn,n | → 1 as n → ∞. For the sum with i = 0, it suffices to estimate the following by Lemma 2.9, tn≤j≤n |S j−ρn+R (G)| · |S n−j (H)| |A tn,n | ≺ 1 exp(ν G (ρn − R)) tn≤j≤n 1 exp(ν G (n − j)) , which tends to 0, as (ρn − R) → ∞.…”

“…In [7], Duchin-Lelièvre-Mooney proved that Z d for d ≥ 2 is not statistically hyperbolic for any finite generating set. It was also discovered by Duchin-Mooney in [8] that the integer Heisenberg group with any finite generating set is not statistically hyperbolic.…”

The lattice point counting problem on the Heisenberg groups

“…We note further that there has been considerable recent interest in geometric group theory in specific lattice point counting results in the Heisenberg groups. These pertain to Carnot-Carathéodory distances arising from word metrics on the lattice subgroup, and we refer to [1], [7], and the references therein for more on this topic. These counting problems are completely different from those we will consider in the present paper.…”