We study the topology of the space $$\Omega_{p}$$ of admissible paths between two points $$e$$ (the origin) and $$p$$ on a step-two Carnot group $$G$$:
$$\Omega_{p} = \{\gamma \colon I \to G | \gamma$$ admissible, $$\gamma(0) = e, \gamma(1) = p\}$$.
As it turns out, $$\Omega_{p}$$ is homotopy equivalent to an infinite dimensional sphere and in particular it is contractible. The energy function:
$$J \colon \Omega_{p} \to \mathbb{R}$$
is defined by $$J(\gamma) = \frac{1}{2} \int_I {\Vert \dot{\gamma} \Vert}^2$$;critical points of this function are sub-Riemannian geodesics between $$e$$ and $$p$$. We study the asymptotic of the number of geodesics and the topology of the sublevel sets:
$${{\Omega}^s}_p = \{\gamma \in \Omega_{p} \| J(\gamma) \leq s\}$$ as $$s \to \infty$$.
If p is not a vertical point in $$G$$, the number of geodesics joining $$e$$ and $$p$$ is bounded and the homology of $${{\Omega}^s}_p$$ stabilizes to zero for $$s$$ large enough.
A completely different behavior is experienced for the generic vertical $$p$$. In this case we show that $$J$$ is a Morse-Bott function: geodesics appear in isolated families (critical manifolds), indexed by their energy. Denoting by $$l$$ the corank of the horizontal distribution on $$G$$, we prove that:
Card{Critical manifolds with energy less than $$s$$} $$ \leq O(s)^l$$.
Despite this evidence, Morse-Bott inequalities $$b({{\Omega}^s}_p) \leq O(s)^l$$ are far from being sharp and we show that the following stronger estimate holds:
$$b{{\Omega}^s}_p \leq O(s)^{l-1}$$. Thus each single Betti number $$b_i ({{\Omega}^s}_p) (i > 0)$$ becomes eventually zero as $$s \to \infty$$, but the sum of all of them can possibly increase as fast as $$O(s)^{l-1}$$. In the case $$l = 2$$ we show that indeed
$$b{{\Omega}^s}_p = \tau (p)s + o(s) (l = 2).$$ The leading order coefficient $$\tau(p)$$ can be analytically computed using the structure constants of the Lie algebra of $$G$$.
Using a dilation procedure, reminiscent to the rescaling for Gromov-Hausdorff limits, we interpret these results as giving some local information on the geometry of $$G$$ (e.g. we derive for $$l = 2$$ the rate of growth of the number of geodesics with bounded energy as $$p$$ approaches $$e$$ along a vertical direction).