Notes on nilpotent groups and subriemannian manifolds

Some examples of nilpotent groups and Lie algebras:

The Heisenberg group

\[\mathfrak{h} = \langle X, Y, Z \mid [X,Y] = Z, \text{all other pairs commute}\rangle\]

The Engel group

\[\mathfrak{e} = \langle X, Y, Z, W \mid [X,Y] = Z, [X,Z] = W, \text{all other pairs commute}\rangle\] This is a step–3 nilpotent group.

Filiform nilpotent Lie algebras

A nilpotent Lie algebra is filiform if the dimension is equal to the step plus \(1\); the Heisenberg group and the Engel group are both filiform. For \(k>1\), one filiform Lie algebra of dimension \(k\) is given by: \[\mathfrak{g}_k = \langle e_1,\dots, e_k\mid [e_i,e_j] = (j-i) e_{i+j} \text{ for all } i+j\le k\rangle;\] (Exercise: Check that this satsifies the Jacobi identity.)

Free nilpotent groups and Lie algebras

Let \(F_r\) be the free group on \(r\) generators, \(F_r=\langle g_1,\dots, g_r\rangle\) and let \(k\ge 1\). The free step--\(k\) nilpotent group of rank \(r\) is the group \(F_{r,k} := F_r/(F_r)_{k+1}\); any step--\(k\) nilpotent group with at most \(r\) generators is a quotient of \(F_{r,k}\).

Notably, the group \(F_{2,2}\) is isomorphic to the Heisenberg group.

The easiest way to describe this group is to embed it as a lattice in a nilpotent Lie group. Let \(\mathfrak{f}_r\) be the free Lie algebra of rank \(r\); then \(F_{r,k}\) is a lattice in the Lie group corresponding to \(\mathfrak{f}_{r,k} = \mathfrak{f}_r/(\mathfrak{f}_r)_{k+1}\).

One can describe \(\mathfrak{f}_r\) combinatorially; if \(e_1,\dots, e_r\) are the generators of \(\mathfrak{r}\), then each quotient \((\mathfrak{f}_r)_n/(\mathfrak{f}_r)_{n+1}\) is an abelian Lie algebra spanned by iterated brackets of the \(e_i\)'s. Because of the Jacobi identity, these iterated brackets are not linearly independent; Witt showed that \(\mathfrak{f}_{r,k}/\mathfrak{f}_{r,k+1}\) has a basis consisting of \[M_r(k) = \frac{1}{k} \sum_{d\mid k} \mu(d) r^{k/d}\] iterated brackets, where \(\mu\) is the Möbius function.

The geometry of \(\mathbb{H}\) is a good starting point for understanding the geometry of nilpotent groups in general.

We represent \(\HH\) as \(\R^3\) equipped with the multiplication \[(x,y,z)\cdot(x',y',z') = \left(x+x', y+y', z + z' + \frac{xy'-yx'}{2}\right).\] (This is a little different from the coordinates we used before – this is the multiplication on \(\mathfrak{h}\) coming from the Baker–Campbell–Hausdorff formula.)

We define a left-invariant metric on \(\HH\): the standard basis of \(\R^3\) extends to three left-invariant vector fields

\begin{align*} X & = (1,0,-\tfrac{y}{2}) \\ Y & = (0,1,\tfrac{x}{2}) \\ Z & = (0,0,1). \end{align*}

We define a Riemannian metric so that these fields are orthogonal: \[dg^2 = dx^2 + dy^2 + \left(dz + \frac{y}{2} dx - \frac{x}{2} dy\right);\] this metric is left-invariant.

Since this is a Riemannian metric, we can calculate its geodesics: it turns out that the geodesics are based on helices. For example, suppose that \(\gamma=(\gamma_x,\gamma_y,\gamma_z)\) is a path from \(0\) to \((0,0,C)\) for \(C>0\). We will find conditions such that \(\gamma\) is length-minimizing.

We have \[\ell(\gamma) = \int_0^1 \sqrt{(\gamma_x')^2 + (\gamma_y')^2 + (\gamma_z' - \tfrac{\gamma_x\gamma_y'}{2} + \tfrac{\gamma_y\gamma_x'}{2})^2}\; dt.\]

We analyze this by defining two auxiliary functions. Let \(\pi\from \HH \to \R^2\) be projection to the \(xy\)–plane and let \[L(t) = \ell(\pi(\gamma([0,t]))) = \int_0^t \sqrt{(\gamma_x')^2 - (\gamma_y')^2}\; dt\] \[A(t) = \frac{1}{2} \int_0^t \gamma_y\gamma_x' - \gamma_x\gamma_y'\; dt.\] Note that \(A(t)\) is the signed area of the closed curve formed by joining the endpoints of \(\pi(\gamma([0,t]))\) by a line segment. Let \(L_\gamma =L(1)\) be the length of \(\pi \circ \gamma\) and let \(A_\gamma =A(1)\) be the signed area of \(\pi \circ \gamma\) (which is a closed curve).

Then \[\ell(\gamma) = \int_0^1 \sqrt{(L')^2 + (\gamma_z' - A')^2}\; dt.\] Consider the curve \(g(t) = (L(t), \gamma_z(t) - A(t))\). Then \(g\) connects \((0,0)\) to \((0,C-A_\gamma)\) and \(\ell(\gamma) = \ell(g)\) — in particular, we have \[\ell(\gamma) \ge \sqrt{L^2 + (C-A_\gamma)^2},\] and we can achieve \(\ell(\gamma) = \sqrt{L_\gamma^2 + (C-A_\gamma)^2}\) by reparametrizing \(\gamma_x\) and \(\gamma_y\) and adjusting \(\gamma_z\) so that \(g\) is a line segment.

So we want to minimize \(f(\gamma) = L_\gamma^2 + (C-A_\gamma)^2\) over closed curves \(\gamma\). Suppose \(A_\gamma = \delta\). By the isoperimetric inequality for the plane, \(L_\gamma \ge \sqrt{2\pi\delta}\), and \[f(\gamma) \ge 2\pi\delta + (C-\delta)^2,\] and this is sharp if \(\gamma\) is a circle. We minimize over \(\delta\) to find that if \(\gamma\) is a geodesic, then

• if \(0\le C\le \pi\), then \(\gamma\) is a vertical line segment
• if \(C > \pi\), then \(\pi \circ \gamma\) is a circle of area \(C - \pi\).

We're interested in large-scale geometry, so what does this metric look like on the large scale? We construct the scaling limit or the asymptotic cone by letting \(d_g\) be the Riemannian distance function and considering the spaces \((\HH, R^{-1} d_g)\) – the unit ball in \((\HH, R^{-1} d_g)\) corresponds to the ball of radius \(R\) in \((\HH, d_g)\).

We first define a sequence of scalings of \(\HH\). For \(t>0\), let \(s_t \from \HH \to \HH\), \((x,y,z)\mapsto (tx,ty,t^2z)\). This is a Lie group isomorphism, and from our calculations with the discrete group, we expect that \[R^{-1} d_g(0,s_{R}(x,y,z)) = R^{-1} d_g(0,(R x,R y,R^2 z)) \approx R^{-1} \max\{|R x|, |R y|, \sqrt{|R^2 z|}\}\] stays bounded as \(R^{-1} \to 0\).

We can calculate the limit by considering the pullback

\begin{align*} dg_R^2 & = R^{-2} s_{R}^*(dg^2)\\ & = R^{-2}\left[(R dx)^2 + (R dy)^2 + \left(R^2 dz + \frac{R y}{2} R dx - \frac{R x}{2} R dy\right)^2\right] \\ & = dx^2 + dy^2 + R^2 \left(dz + \frac{y}{2} dx - \frac{x}{2} dy\right)^2. \end{align*}

This is the same left-invariant metric we saw before – except now the \(Z\) field has length \(R\). As \(R\to \infty\), the length of \(Z\) goes to \(\infty\).

The only vectors whose length stays bounded are the vectors in \[V_1 = \ker(dz + \frac{y}{2} dx - \frac{x}{2} dy) = \langle X, Y\rangle.\] We call these horizontal vectors, and we call curves tangent to \(V_1\) horizontal curves.

• Definition
• Examples and non-examples
• Scaling automorphisms
• The scale-invariant sub-riemannian metric \(d_\infty\)

The fact that the scaling automorphisms of a Carnot group scale the subriemannian metric leads to an estimate of the subriemannian distance.

Let \(\mathfrak{g}\) be Carnot and let \(X_1,\dots, X_m \in \mathfrak{g}\) be a basis such that there are \(j_i\) satisfying \[V_i = \langle X_{j_i}, \dots, X_{j_{i+1}-1} \rangle.\] For each \(i\), let \(d_i\) be such that \(V_i\in X_{d_i}\). We parameterize \(G\) by setting \((a_1,\dots, a_m) = \exp(\sum a_i X_i)\). Then the following theorem holds:

Consequently, \(\vol B(0,R;d_\infty) \approx R^Q\), where \(Q = \sum d_i = \sum i \dim(V_i)\).

The proof follows by scaling; there is an \(\epsilon>0\) such that if \(\rho(v)\le \epsilon\), then \(v\in B(0,1;d_\infty)\) and if \(v\in B(0,1;d_\infty)\), then \(\rho(v)<\epsilon^{-1}\). Then for any \(v\in G\), we have \(s_{\epsilon \rho(v)^{-1}}(v)\in B(0,1;d_\infty)\) and thus \(v\in B(0, \epsilon^{-1}\rho(v))\); likewise, \(v\not \in B(0, \epsilon\rho(v))\).

While Carnot groups have scaling automorphisms and can be given a scale-invariant metric, an arbitrary nilpotent Lie group might not have scalings. Nonetheless, we can describe the geometry using the lower central series and the associated graded algebra.

Let \(G\) be a nilpotent Lie group of step \(k\) and let \(\mathfrak{g}_i\) be the lower central series of \(\mathfrak{g}\). An adapted basis of \(\mathfrak{g}\) is a basis \(X_1,\dots, X_m\) such that for each \(i\), there is a \(j_i\) satisfying \(\mathfrak{g}_i = \langle X_{j_i},\dots, X_m\rangle\). Let \((a_1,\dots,a_m) = \exp(\sum a_i X_i)\). We can use this basis to state the ball-box theorem for \(G\).

(We omit the proof; for full details, see one of the sources in the further reading section of the website.)

We can describe the geometry in finer detail using the associated graded algebra \(\mathfrak{g}_\infty\).

Let \(V_i=\mathfrak{g}_i/\mathfrak{g}_{i+1}\). We note that \(V_i\) is abelian and that, by the Jacobi identity, \([\mathfrak{g}_i, \mathfrak{g}_j] \subset \mathfrak{g}_{i+j}\).

Let \[\mathfrak{g}_\infty = V_1\oplus\dots\oplus V_k.\] For any \(v_i \in \mathfrak{g}_{i}\) and \(v_j\in \mathfrak{g}_{j}\), \[[v_i + \mathfrak{g}_{i+1}, v_j + \mathfrak{g}_{j+1}]\subset [v_i,v_j] + \mathfrak{g}_{i+j+1},\] so the bracket on \(\mathfrak{g}\) induces a well-defined bracket on \(\mathfrak{g}_\infty\), which makes \(\mathfrak{g}_\infty\) Carnot – we call \(\mathfrak{g}_\infty\) the associated graded algebra of \(\mathfrak{g}\).

Pansu proved the following theorem:

This convergence is easiest to see from an algebraic standpoint. Since \(X_1,\dots, X_m\) is an adapted basis, the subset \(X_{j_i},\dots, X_{j_{i+1}-1}\) is a basis for \(V_i=\mathfrak{g}_i/\mathfrak{g}_{i+1}\). This lets us identify (the vector spaces) \(\mathfrak{g}\) and \(\mathfrak{g}_\infty\) with (the vector space) \(\R^m\). For \(v,w\in \R^m\), we write \([v,w]\) and \([v,w]_\infty\) for the brackets on \(\R^m\) coming from the identification with \(\mathfrak{g}\) and \(\mathfrak{g}_\infty\) respectively.

Let \(d_i\) as above, so that \(X_i\in \mathfrak{g}_{d_i}\setminus \mathfrak{g}_{d_i+1}\). We can describe the bracket on \(\mathfrak{g}\) in terms of the structure coefficients \(c_{ij}^k\), i.e., the real numbers such that \[[X_i,X_j] = \sum_k c_{ij}^k X_k.\] Note that \([X_i,X_j]\in \mathfrak{g}_{d_i+d_j}\) and that \(\mathfrak{g}_{d_i+d_j}\) is generated by the \(X_k\)'s such that \(d_k \ge d_i+d_j\). Therefore, \(c_{ij}^k = 0\) unless \(d_k \ge d_i+d_j\), and \[[X_i,X_j] = \sum_{d_k\ge d_i+d_j} c_{ij}^k X_k.\]

We can describe the bracket on \(\mathfrak{g}_\infty\) similarly. Recall that \[[X_i,X_j]_\infty = [X_i,X_j]+\mathfrak{g}_{d_i+d_j+1}\in V_{d_i+d_j}.\] Since \(X_k\in \mathfrak{g}_{d_k}\), the formula above becomes \[[X_i,X_j]_\infty = \sum_{d_k=d_i+d_j} c_{ij}^k X_k.\] That is, \([,]_\infty\) ignores the parts of \([,]\) that lie in the wrong \(V_k\).

We claim that rescalings of \(\mathfrak{g}\) converge to \(\mathfrak{g}_\infty\) in the following sense:

We write \(G\) and \(G_\infty\) as two different group operations on \(\R^m\): for \(v,w\in \R^m\), let \[v\cdot w = v + w + \frac{1}{2}[v,w]+\dots\] \[v\ast w = v + w + \frac{1}{2}[v,w]_\infty+\dots\] By the Baker-Campbell-Hausdorff formula, the lemma implies \[\lim_{R\to \infty} s_R^{-1}(s_R(v)s_R(w)) = v\ast w.\] That is, the group operation on \(G\) can be approximated by the group operation on \(G_\infty\) on large balls.

• If \(G\) is nilpotent and \(V\) is a left-invariant subbundle of the tangent bundle such that \(V(0)\) is isomorphic to \(\mathfrak{g}/[\mathfrak{g}, \mathfrak{g}]\), then there's a scaling map \(s_t\) so that rescalings \((s_R^{-1})_*(V)\) converges to the horizontal bundle of \(G_\infty\).
• Thus, if \(G\) is given a Riemannian metric and \(\gamma_R\from [0,1]\to G\) is constant-speed with \(\ell(\gamma_R)=R\), then \(s_{R^{-1}}(\gamma_R)\) is close to a horizontal curve with subunit speed in \(G_\infty\).
• Therefore, if \(\gamma\) is a geodesic from \(s_R(v)\) to \(s_R(w)\), then there's a horizontal curve of length \(\ell(\gamma)/R\) that connects \(v\) to \(w'\), where \(w'\) has coordinates close to \(w\).
• By the Ball-Box theorem, \(w'\) is metrically close to \(w\). Taking these arguments further, we get \[\lim_{R\to \infty} R^{-1}d_G(s_R(v),s_R(w)) = d_{G_\infty}(v,w).\]
• Similar results apply to equiregular subRiemannian manifolds, in which case, by a theorem of Mitchell, scalings of the balls \(B_\epsilon(p)\) converge to a Carnot group as \(\epsilon\to 0\). In order to prove this, one proves that there are privileged coordinates for which the ball-box theorem holds. Using these coordinates, one can define a scaling so that scalings of the horizontal distribution approach a limit. Finally, one proves that this limit is the horizontal distribution of a Carnot group.

The scaling limits of nilpotent groups are crucial to understanding quasi-isometries between nilpotent groups. If \(G\) and \(H\) are nilpotent Lie groups with Riemannian metrics and \(f\from G\to H\) is a quasi-isometry, then scalings of \(f\) converge to a bilipschitz map \(f_\infty\from G_\infty \to H_\infty\).

We can then apply the following theorem of Pansu:

Before proving the theorem, we first prove a special case.

Recall some facts from algebraic topology:

• Let \(\Gamma\) be a discrete group. An Eilenberg–MacLane space of type \(K(\Gamma,1)\) is a CW complex \(X\) with \(\pi_1(X)=\Gamma\) and \(\pi_n(X)=0\) for \(n > 1\). Equivalently, a \(K(\Gamma,1)\) is a CW complex with \(\pi_1(X)=\Gamma\) such that the universal cover is contractible. Any two \(K(\Gamma,1)\)'s are homotopy equivalent.
• Given a group \(\Gamma\), we define \(H^n(\Gamma;\R) = H^n(K(\Gamma,1);\R)\) for any \(K(\Gamma,1)\). Since any two \(K(\Gamma,1)\)'s are homotopy equivalent, this doesn't depend on the choice of space.

When \(\Gamma\) is a lattice in a nilpotent group, we can describe the cohomology of \(\Gamma\) in terms of the Lie algebra.

Here, \(H^*(\mathfrak{g})\) is the cohomology of the complex \(C^*(\mathfrak{g})\) of left-invariant differential forms on \(G\). Any such form is determined by its value at \(0\), so we can write this complex as: \[\R \to \mathfrak{g}^* \to \wedge^2 \mathfrak{g}^* \to \dots \to \wedge^n \mathfrak{g}^*.\] If \(\omega\in \wedge^d \mathfrak{g}^*\) and \(\Sigma\) is an \(d\)–dimensional surface in \(G\), we define \[\omega(\Sigma) = \int_{\Sigma} \omega.\]

The differential on \(C^*(\mathfrak{g})\) can be written in terms of the Lie bracket. For \(\omega\in \mathfrak{g}^*\), \[d\omega(v,w) = \omega([v,w]).\]

A locally compact group \(G\) is amenable if there is a finitely-additive left-invariant probability measure on \(G\). That is, there is a function \(\mu\from 2^G\to [0,1]\) such that:

• \(\mu(G)=1\)
• for \(g\in G\), \(\mu(gS)=\mu(S)\)
• if \(S,T\subset G\) and \(S\cap T=\emptyset\), then \(\mu(S\cup T) = \mu(S) + \mu(T)\)

Roughly, \(\mu\) is a way to measure how big a subset of \(G\) is. For example, when \(G=\Z\) and \(a>0\), this implies \(\mu(a\Z) = \frac{1}{a}\), because \(\mu(a\Z)=\mu(a\Z+b)\) for any \(b\) by left-invariance, and \[\mu(\Z)=\mu(a\Z) + \mu(a\Z+1) + \dots + \mu(a\Z+a-1).\] Equivalently, an invariant mean can be viewed as a left-invariant functional \(\mu\from L_\infty(G)\to \R\) such that:

• if \(f\ge 0\), then \(\mu(f)\ge 0\)
• \(\mu\) has norm \(1\).

It's not obvious how to construct a function like this, and in fact, the construction uses the axiom of choice, in the form of the Hahn–Banach theorem. We first construct an ultralimit. Let \(V\subset \ell_\infty(\N)\) be the set of convergent subsequences. Then \(\lim\from V\to \R\) is a linear map with norm \(1\). By Hahn–Banach, it has a linear extension \(\underline{\lim}\from \ell_\infty(\N)\to \R\), also with norm \(1\); we think of \(\underline{\lim}\) as a way to assign a limit to any bounded sequence.

Next, we define a Følner sequence for a group. This is a sequence of subsets of the group with small boundary. Let \(G\) be a group and let \(K\) be a finite generating set. For \(S\subset G\), let \(\partial S = \{g\in S\mid gK\not\subset S\}\). A sequence of nonempty finite sets \(S_1,\dots\) is a Følner sequence if \(\frac{\# \partial S_i}{\#S_i}\to 0\). If a group has polynomial growth, then the metric balls \(B_n\) form a Følner sequence, but many groups with nonpolynomial growth also have Følner sequences, including all solvable groups.

We can use a Følner sequence for \(\Gamma\) to construct an invariant mean. Let \(S_1,\dots\subset \Gamma\) be a Følner sequence. For \(T\subset \Gamma\), let \[\mu(T) = \underline{\lim} \frac{|T\cap S_n^{-1}|}{|S_n|}.\] Since \(\frac{|T\cap S_n^{-1}|}{|S_n|}\in [0,1]\), this limit is between \(0\) and \(1\). It satisfies \(\mu(\Gamma)=1\) and, by the linearity of \(\underline{\lim}\), \(\mu\) is finitely additive.

To show left-invariance, let \(k\) be a generator of \(\Gamma\). Then \[|kT\cap S_n^{-1}| = |T\cap k^{-1} S_n^{-1}| = |T\cap (k S_n)^{-1}|;\] it follows that \[\big||kT\cap S_n^{-1}| - |T\cap S_n^{-1}|\big| \le |k S_n\setminus S_n| \le |\partial S_n|.\] Therefore, \[\underline{\lim} \frac{|T\cap S_n^{-1}|}{|S_n|} - \frac{|kT\cap S_n^{-1}|}{|S_n|} = 0\] and \(\mu(T) = \mu(kT)\). Since this holds for every generator of \(\Gamma\), we have \(\mu(gT)=\mu(T)\) for all \(g\in \Gamma\).

Let \(\Lambda\) and \(\Gamma\) be torsion-free and let \(f\from \Lambda \to \Gamma\) be a quasi-isometry. Let \(\Delta(\Lambda)\) be the complete simplex with vertex set \(\Lambda\). Then we can write \(H^*(\Lambda;\R)\) as the cohomology of the complex of invariant cocycles, \[C^n(\Lambda\backslash \Delta(\Lambda);\R) = \{\omega\from \Lambda^{n+1}\to \R \mid \omega \text{ is alternating and left-invariant}\}.\]

For \(\omega\in C^n(\Lambda\backslash \Delta(\Lambda);\R)\) and \(D>0\), let \(f^*\omega\in C^n(\Gamma;\R)\), \[f^*\omega(\gamma_0,\dots,\gamma_n) = \omega(f(\gamma_0),\dots, f(\gamma_n)).\] Note that \(f^*\omega\) is generally not invariant, but we can average over translates to construct an invariant cocycle. \[M_\omega(D) = \max_{\diam\{\lambda_0,\dots, \lambda_n\}\le D} |\omega(\lambda_0,\dots, \lambda_n)|;\] since \(\omega\) is left-invariant, this maximum is finite. Since \(f\) is a quasi-isometry, there is a \(C>0\) such that for any \(\omega\in C^n(\Lambda\backslash \Delta(\Lambda);\R)\) and any \(\gamma_0, \dots, \gamma_n, H\in \Gamma\), \[|f^*\omega(h\gamma_0,\dots,g\gamma_n)| = |\omega(f(h \gamma_0), \dots, f(h\gamma_n))| \le M_\omega(C\diam \{\lambda_0, \dots, \lambda_n\}+C).\] In particular, if \(\mu\) is an invariant mean on \(\Lambda\), we can define an invariant cocycle \[\overline{f^*}\omega(\gamma_0,\dots, \gamma_n) = \int_\Gamma f^*\omega(h\gamma_0,\dots,h\gamma_n)\ud \mu.\] One can check that \(\overline{f^*}(d\omega) = d(\overline{f^*}\omega)\), so this induces a map \(\overline{f^*}\from H^n(\Gamma;\R)\to H^n(\Lambda;\R)\) – what can we say about this map?