Finite-energy functions on a Cayley complex

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

Let \(\Gamma\) be a group with a finite presentation. A real function \(F\) on \(\Gamma\) has an energy: the sum, over the edges of a Cayley graph, of the squares of the differences of \(F\) across the edges. The functions of finite energy form a Hilbert space \(\mathcal D\) of gradients. It contains the closed subspace \(\mathcal D_0\) spanned by the gradients of finitely supported functions, and the question whether \(\mathcal D_0=\mathcal D\) is a question about the group. For \(\mathbb Z\) the answer is yes; for a free group on two generators it is no (Exercises 5.2 and 5.3).

This lesson treats groups with \(\mathcal D_0\ne\mathcal D\). Its main result (Theorem 4.1) produces one particular function \(f:\Gamma\to[0,1]\) of finite energy that does not tend to a constant at infinity and whose level sides \(\{f<t\}\) and \(\{f>t\}\) are connected subgraphs for every level \(t\) that is not a value of \(f\); moreover the set of edges crossing such a level is connected in the sense appropriate to the \(2\)-cells of the Cayley complex. Lemma 3.1 bounds how often the crossing sets of two finite-energy functions meet. These are the two inputs for the construction of a boundary of \(\Gamma\) from finite-energy functions in OpenAI's proof of the Singer conjecture in dimension four outside the middle degree [OpenAI-S, Section 3].

The quotient \(\mathcal D/\mathcal D_0\) is the reduced \(\ell^2\)-cohomology of \(\Gamma\) in degree one; this name is not used below.

We assume the core courses Functional Analysis (closed subspaces of a Hilbert space, the Riesz representation theorem) and Measure and Integration (D10) (Tonelli's theorem, dominated convergence), and the notions of free group and presentation of a group.

1. The Cayley complex of a triangular presentation

Let \(S\) be a finite set and \(R\) a finite set of words in the letters \(s\) and \(s^{-1}\), \(s\in S\), each of length \(1\), \(2\) or \(3\). The presentation \(\langle S\mid R\rangle\) is then called triangular. Let \[ \Gamma=\langle S\mid R\rangle \] be the group it presents: the quotient of the free group on \(S\) by the normal subgroup generated by \(R\). Every finitely presented group has a triangular presentation (Exercise 5.1).

Vertices and edges. The Cayley graph \(G\) has vertex set \(\Gamma\) and edge set \(E=\Gamma\times S\). The edge \(e=(g,s)\) has initial vertex \(e^-=g\) and terminal vertex \(e^+=gs\). Loops and parallel edges occur if some generator is trivial in \(\Gamma\) or two generators are equal in \(\Gamma\); they are kept. Every vertex lies on at most \(2|S|\) edges, and \(G\) is connected because \(S\) generates \(\Gamma\). A set of vertices is connected if any two of its vertices are joined by a path of edges through vertices of the set.

Paths. A word \(w=x_1\cdots x_k\) in the letters \(s^{\pm1}\) and a vertex \(g\) determine a path with vertices \(g,\ gx_1,\ gx_1x_2,\dots\); the letter \(x_i=s\) traverses the edge \((gx_1\cdots x_{i-1},s)\) forwards, and the letter \(x_i=s^{-1}\) traverses the edge \((gx_1\cdots x_i,s)\) backwards.

Cells. A cell is a pair \((g,r)\in\Gamma\times R\). Its boundary is the path of the word \(r\) from \(g\), which is closed because \(r\) is trivial in \(\Gamma\); the edges on this path are the edges of the cell. A cell has at most three edges, and an edge lies in at most \(3|R|\) cells.

Cochains. Let \(A\) be an abelian group. A \(1\)-cochain is a function \(c:E\to A\). Its value on a traversal is \(c(e)\) if the edge \(e\) is traversed forwards and \(-c(e)\) if backwards, and its integral \(\int_{g,w}c\) along the path of \(w\) from \(g\) is the sum of its values on the traversals. A cochain is a cocycle if its integral around the boundary of every cell is zero. The gradient of \(\varphi:\Gamma\to A\) is the cochain \[ d\varphi(e)=\varphi(e^+)-\varphi(e^-). \] The integral of \(d\varphi\) along a path is the difference of the values of \(\varphi\) at its end and its beginning, so every gradient is a cocycle.

Lemma 1.1 (cocycles are gradients). Every cocycle \(c:E\to A\) is the gradient of a function \(\varphi:\Gamma\to A\), and \(\varphi\) is unique up to an additive constant.

Proof. Integrals along closed paths vanish. Let \(u\) be a word that is trivial in \(\Gamma\) and \(h\) a vertex. In the free group, \(u\) equals a product of conjugates \(vr^{\pm1}v^{-1}\) with \(r\in R\). So the word \(u\) is obtained from a concatenation of words \(vr^{\pm1}v^{-1}\) by inserting and deleting pairs \(xx^{-1}\). Three remarks. The integral is additive under concatenation. Inserting or deleting \(xx^{-1}\) does not change it, because the two traversals are of the same edge in opposite directions. And along \(vrv^{-1}\) from \(h'\) it equals \[ \begin{aligned} &\int_{h',v}c+\int_{h'v,r}c+\int_{h'v,v^{-1}}c\\ &\qquad=\int_{h'v,r}c=0, \end{aligned} \] because the path of \(v^{-1}\) from \(h'v\) is the path of \(v\) from \(h'\) run backwards, and \(c\) is a cocycle; the same holds with \(r^{-1}\), whose path is that of \(r\) run backwards. Hence \(\int_{h,u}c=0\).

The function. For \(g\in\Gamma\) choose a word \(w\) representing \(g\) and put \(\varphi(g)=\int_{1,w}c\). If \(w'\) also represents \(g\), then \(w^{-1}w'\) is trivial in \(\Gamma\), and the integral along \(ww^{-1}w'\) from \(1\) equals both \(\int_{1,w'}c\) (delete pairs) and \(\int_{1,w}c+\int_{g,w^{-1}w'}c=\int_{1,w}c\). So \(\varphi\) is well defined. For an edge \((g,s)\), the word \(ws\) represents \(gs\), and \(\varphi(gs)-\varphi(g)=c(g,s)\).

Uniqueness. If \(d\varphi=d\varphi'\), then \(\varphi-\varphi'\) has the same value at the two ends of every edge, so it is constant, since \(G\) is connected. \(\blacksquare\)

In topological terms the lemma says that the Cayley complex is simply connected; only the lemma is used.

2. Energy

For \(F:\Gamma\to\mathbb R\) the energy is \[ \mathcal E(F)=\sum_{e\in E}|dF(e)|^2\in[0,\infty]. \] Let \(\ell^2(E)\) be the real Hilbert space of square summable functions on \(E\), with norm \(\|\cdot\|\), and \[ \mathcal D=\{dF:\ \mathcal E(F)<\infty\}\subseteq\ell^2(E). \] Let \(\mathcal D_0\) be the closure in \(\ell^2(E)\) of the set of gradients \(du\) of functions \(u\) with finite support; such gradients have finite support, so they lie in \(\mathcal D\).

Lemma 2.1. \(\mathcal D\) is a closed subspace of \(\ell^2(E)\), and \(\mathcal D_0\subseteq\mathcal D\).

Proof. By Lemma 1.1, \(\mathcal D\) is the set of cocycles in \(\ell^2(E)\). For each cell the integral around its boundary is a sum of at most three coordinates with signs, a continuous linear functional on \(\ell^2(E)\). So \(\mathcal D\) is an intersection of closed subspaces. The second statement follows because \(\mathcal D\) is closed. \(\blacksquare\)

A function \(F:\Gamma\to\mathbb R\) tends to the constant \(c\) at infinity if for every \(\varepsilon>0\) the set \(\{v:|F(v)-c|\ge\varepsilon\}\) is finite.

Lemma 2.2 (truncation). Let \(\theta:\mathbb R\to\mathbb R\) satisfy \(|\theta(x)-\theta(y)|\le|x-y|\). Then \(|d(\theta\circ F)(e)|\le|dF(e)|\) for every edge, by the inequality for \(\theta\) at the two ends of \(e\), and so \(\mathcal E(\theta\circ F)\le\mathcal E(F)\). This applies to \(\theta(x)=\min(1,\max(0,x))\) and to the truncation \(\theta_\varepsilon(x)=\min(\varepsilon,\max(-\varepsilon,x))\).

Lemma 2.3. If \(F\) has finite energy and tends to a constant at infinity, then \(dF\in\mathcal D_0\).

Proof. Let \(c\) be the constant and \(H=F-c\), so \(dH=dF\). For \(\varepsilon>0\) put \(u_\varepsilon=H-\theta_\varepsilon\circ H\). This function vanishes where \(|H|\le\varepsilon\), so it has finite support. By Lemma 2.2, \[ \begin{aligned} |dF(e)-du_\varepsilon(e)|&=|d(\theta_\varepsilon\circ H)(e)|\\ &\le\min\bigl(|dF(e)|,2\varepsilon\bigr). \end{aligned} \] The right side is dominated by \(|dF(e)|\), which is square summable, and tends to \(0\) for each \(e\) as \(\varepsilon\to0\). By dominated convergence \(\|dF-du_\varepsilon\|\to0\). \(\blacksquare\)

So \(\mathcal D_0\ne\mathcal D\) can only happen through functions that do not settle down at infinity. If \(\Gamma\) is finite, all functions have finite support and \(\mathcal D_0=\mathcal D\).

3. Cuts

Let \(F:\Gamma\to\mathbb R\) and \(t\in\mathbb R\). The cut of \(F\) at level \(t\) is the set of edges that cross the level: \[ \begin{gathered} S(F,t)=\{e\in E:\ m_e<t<M_e\},\\ m_e=\min(F(e^-),F(e^+)),\\ M_e=\max(F(e^-),F(e^+)). \end{gathered} \] The level \(t\) is regular for \(F\) if \(t\) is not a value of \(F\). For a regular level every vertex lies in exactly one of the level sides \(\{F<t\}\) and \(\{F>t\}\), and \(S(F,t)\) is the set of edges with one end in each side.

Write \(e\sim e'\) for two edges if \(e=e'\) or \(e\) and \(e'\) are edges of a common cell. For a given \(e\) there are at most \[ D=1+9|R| \] edges \(e'\) with \(e\sim e'\), since \(e\) lies in at most \(3|R|\) cells with at most three edges each. Two edges of a set \(C\subseteq E\) are dually adjacent if they are edges of a common cell; \(C\) is dually connected if any two of its edges are joined by a finite sequence of edges of \(C\) in which consecutive ones are dually adjacent. The classes of the equivalence relation generated on \(C\) by dual adjacency are the dual components of \(C\).

Lemma 3.1 (two cuts rarely interact). Let \(F,F'\) have finite energy, and for \(s,t\in\mathbb R\) let \(N(s,t)\) be the number of pairs \((e,e')\) with \[ \begin{gathered} e\sim e',\\ e\in S(F,s),\qquad e'\in S(F',t). \end{gathered} \] Then \[ \int_{\mathbb R^2}N(s,t)\,ds\,dt\le D\,\|dF\|\,\|dF'\| . \] In particular \(N(s,t)<\infty\) for almost every \((s,t)\). The case \(F=F'\) is included.

Proof. The set of \(s\) with \(e\in S(F,s)\) is an open interval of length \(|dF(e)|\). So by Tonelli's theorem, for the product of Lebesgue measure on \(\mathbb R^2\) with the counting measure on pairs, \[ \begin{aligned} &\int_{\mathbb R^2}N(s,t)\,ds\,dt\\ &\qquad=\sum_{e\sim e'}|dF(e)|\,|dF'(e')| . \end{aligned} \] By the Cauchy–Schwarz inequality for sums over the pairs with \(e\sim e'\), the right side is at most \[ \begin{gathered} \Bigl(\sum_{e\sim e'}|dF(e)|^2\Bigr)^{1/2}\\ {}\cdot\Bigl(\sum_{e\sim e'}|dF'(e')|^2\Bigr)^{1/2}, \end{gathered} \] and each factor is at most \(\sqrt D\) times the norm, because every edge occurs in at most \(D\) pairs as first entry and in at most \(D\) pairs as second entry. \(\blacksquare\)

4. A function with connected level sides

Theorem 4.1. Let \(\Gamma=\langle S\mid R\rangle\) be a triangular presentation with \(\mathcal D_0\ne\mathcal D\). There is a function \(f:\Gamma\to[0,1]\) of finite energy with the following properties.

  1. \(df\notin\mathcal D_0\), and \(f\) does not tend to a constant at infinity.
  2. For every level \(t\in(0,1)\) that is regular for \(f\), the level sides \(\{f<t\}\) and \(\{f>t\}\) are nonempty and connected.
  3. For every such \(t\), the cut \(S(f,t)\) is nonempty and dually connected.

Proof. Step 1: point evaluations. Fix the vertex \(a=1\). For a vertex \(b\ne a\) define \[ \begin{gathered} L_b:\mathcal D\to\mathbb R,\\ L_b(dF)=F(b)-F(a). \end{gathered} \] This is well defined, because \(dF\) determines \(F\) up to a constant. It is bounded: along a path from \(a\) to \(b\) with \(\ell\) distinct edges \(e_1,\dots,e_\ell\), \[ \begin{aligned} |F(b)-F(a)|&\le\sum_i|dF(e_i)|\\ &\le\sqrt\ell\,\|dF\| . \end{aligned} \] It is not zero: the indicator function of \(\{b\}\) has finite energy and \(L_b\) of its gradient is \(1\). By Lemma 2.1, \(\mathcal D\) is a Hilbert space, so there is \(r_b\in\mathcal D\) with \(L_b(\omega)=\langle\omega,r_b\rangle\) for all \(\omega\in\mathcal D\).

Step 2: a minimizer outside \(\mathcal D_0\). If \(\omega=dF\in\mathcal D\) is orthogonal to all \(r_b\), then \(F(b)=F(a)\) for all \(b\), so \(\omega=0\). Hence the \(r_b\) span a dense subspace of \(\mathcal D\). Since \(\mathcal D_0\) is a closed subspace different from \(\mathcal D\), some \(r_b\) is not in \(\mathcal D_0\). Fix such a \(b\) and put \[ \omega_b=\frac{r_b}{\|r_b\|^2}. \] Then \(L_b(\omega_b)=1\), and \(\omega_b\) is the unique element of least norm in the closed affine hyperplane \(\{\omega\in\mathcal D:L_b(\omega)=1\}\): for \(\omega\) in this hyperplane, \(\omega-\omega_b\) is orthogonal to \(r_b\), so \(\|\omega\|^2=\|\omega_b\|^2+\|\omega-\omega_b\|^2\). Let \(f\) be the function with \(df=\omega_b\) and \(f(a)=0\). Then \[ f(a)=0,\qquad f(b)=1,\qquad df\notin\mathcal D_0 , \] and \(f\) has the least energy among all functions \(F\) with \(F(b)-F(a)=1\), and is the only such function with this energy and \(F(a)=0\).

Step 3: \(0\le f\le1\). Let \(\tilde f=\min(1,\max(0,f))\). Then \(\tilde f(a)=0\), \(\tilde f(b)=1\), and \(\mathcal E(\tilde f)\le\mathcal E(f)\) by Lemma 2.2. By the uniqueness in Step 2, \(\tilde f=f\).

Step 4: property 1. \(df\notin\mathcal D_0\) by construction, and then Lemma 2.3 shows that \(f\) does not tend to a constant at infinity.

Step 5: property 2. Let \(t\in(0,1)\) be regular. The side \(\{f>t\}\) contains \(b\) and \(\{f<t\}\) contains \(a\). Suppose \(\{f>t\}\) is not connected. Then there is a vertex \(u_0\in\{f>t\}\) that is not joined to \(b\) by a path inside \(\{f>t\}\); let \(U\) be the set of vertices joined to \(u_0\) by a path inside \(\{f>t\}\), so that \(b\notin U\). Define \(f'\) to be \(t\) on \(U\) and \(f\) elsewhere. Then \(f'(a)=0\), because \(f(a)=0<t\), and \(f'(b)=1\). Compare the energies edge by edge.

An edge of the third kind exists: \(G\) is connected, \(U\) is nonempty and \(a\notin U\). So \(\mathcal E(f')<\mathcal E(f)\), while \(f'(b)-f'(a)=1\). This contradicts Step 2. Hence \(\{f>t\}\) is connected. For \(\{f<t\}\) the same argument applies with a set \(U\subseteq\{f<t\}\) not containing \(a\), where now \(f(u)<t<f(v)\).

Step 6: property 3. Let \(t\in(0,1)\) be regular and \(\chi\) the indicator function of \(\{f>t\}\). The gradient \(d\chi\) takes the values \(\pm1\) on \(S(f,t)\) and \(0\) elsewhere. The cut is nonempty, because both sides are nonempty and \(G\) is connected. Let \(C\) be a dual component of \(S(f,t)\), and let \(c\) be the cochain that equals \(d\chi\) on \(C\) and \(0\) on all other edges.

\(c\) is a cocycle. Indeed, the edges of a cell that lie in \(S(f,t)\) are pairwise dually adjacent, so either all of them lie in \(C\) or none does. In the first case the integral of \(c\) around the cell equals that of \(d\chi\), which is zero; in the second case \(c\) vanishes on all edges of the cell.

By Lemma 1.1, \(c=d\varphi\) for some \(\varphi:\Gamma\to\mathbb Z\). An edge with both ends in \(\{f>t\}\) is not in the cut, so \(c\) vanishes on it and \(\varphi\) has the same value at its ends. Since \(\{f>t\}\) is connected, \(\varphi\) is constant on it, with value \(\varphi_+\); likewise \(\varphi\) is constant on \(\{f<t\}\), with value \(\varphi_-\). An edge of \(C\) has one end in each side and \(c=\pm1\) on it, so \(\varphi_+\ne\varphi_-\). Then \(c=d\varphi\) is nonzero on every edge with one end in each side, that is, on every edge of \(S(f,t)\). Since \(c\) vanishes outside \(C\), \(S(f,t)=C\). \(\blacksquare\)

Every level outside \([0,1]\) has an empty cut, and levels \(0\) and \(1\) are not regular. The theorem does not say that \(f\) is the only function with these properties; it depends on the choice of \(b\).

5. Exercises

5.1. Show that every finitely presented group has a triangular presentation.

5.2. Let \(\Gamma=\mathbb Z=\langle s\mid\ \rangle\). Show that \(\mathcal D=\ell^2(E)\) and \(\mathcal D_0=\mathcal D\).

5.3. Let \(\Gamma=\langle s,t\mid\ \rangle\) be the free group on two generators, so that \(G\) is the tree in which every vertex has four neighbours. Let \(F\) be the indicator function of the set of elements whose reduced word begins with \(s\). Show that \(\mathcal E(F)=1\) and \(dF\notin\mathcal D_0\).

5.4. Let \(F\) have finite energy. Show that for almost every \(t\in\mathbb R\) the cut \(S(F,t)\) is finite when the sum \(\sum_e|dF(e)|\) is finite, and give a function of finite energy on \(\mathbb Z\) all of whose cuts at regular levels in \((0,1)\) are infinite.

5.5. In the proof of Theorem 4.1, where is each of the following used: that \(\mathcal D\) is closed; that the level is regular; Lemma 1.1?

6. Solutions

5.1. Let \(\langle S\mid R\rangle\) be a finite presentation and \(r=x_1x_2\cdots x_k\in R\) a relator with \(k>3\). Add a new generator \(y\) and replace \(r\) by the two relators \(y^{-1}x_1x_2\) and \(yx_3\cdots x_k\), of lengths \(3\) and \(k-1\). The new presentation defines the same group: the first relator says \(y=x_1x_2\), and with it the second is equivalent to \(r\); conversely the map sending \(y\) to \(x_1x_2\) is inverse to the obvious map. Repeating this shortens all relators to length at most three. A relator of length \(0\) can be omitted.

5.2. There are no relators, so every cochain is a cocycle and \(\mathcal D=\ell^2(E)\). The edges are \(e_n=(n,s)\) from \(n\) to \(n+1\). Let \(\delta_n\) be the cochain that is \(1\) on \(e_n\) and \(0\) elsewhere; finite combinations of the \(\delta_n\) are dense in \(\ell^2(E)\), so it suffices to show \(\delta_0\in\mathcal D_0\) (the others follow by translation). For \(m\ge1\) let \(u_m\) be the function with \(u_m(n)=0\) for \(n\le0\), \(u_m(n)=1-\frac{n-1}m\) for \(1\le n\le m+1\), and \(u_m(n)=0\) for \(n>m+1\). It has finite support, \(du_m(e_0)=1\), \(du_m(e_n)=-\frac1m\) for \(1\le n\le m\), and \(du_m=0\) on the other edges. So \(\|du_m-\delta_0\|^2=m\cdot\frac1{m^2}\to0\).

5.3. Let \(e_0=(1,s)\). Removing \(e_0\) cuts the tree into the part \(T_+\) of the elements whose reduced word begins with \(s\) and the rest \(T_-\); so \(dF\) is \(1\) on \(e_0\) and \(0\) elsewhere, and \(\mathcal E(F)=1\). Define a cochain \(\theta\) as follows. For \(k\ge1\) there are \(3^k\) edges of \(T_+\) at distance \(k\) from \(e_0\) (joining vertices at distances \(k-1\) and \(k\) from \(s\)); on each, \(\theta\) has absolute value \(3^{-k}\) and the sign for which it is positive when the edge is traversed away from \(e_0\). Symmetrically on \(T_-\), with the sign for which it is positive when the edge is traversed towards \(e_0\). And \(\theta(e_0)=1\). Then \(\|\theta\|^2=1+2\sum_{k\ge1}3^k3^{-2k}<\infty\). At every vertex the total of \(\theta\) over the edges traversed towards the vertex equals the total over the edges traversed away from it: one unit, or \(3^{-k}\), arrives and is split into three equal parts, or three equal parts are joined. For a function \(u\) with finite support, collecting the terms of \(\langle\theta,du\rangle=\sum_e\theta(e)(u(e^+)-u(e^-))\) by vertices gives \(\sum_vu(v)\,n(v)\), where \(n(v)\) is the total arriving at \(v\) minus the total leaving \(v\), which is \(0\). So \(\theta\) is orthogonal to \(\mathcal D_0\), while \(\langle\theta,dF\rangle=\theta(e_0)=1\). Hence \(dF\notin\mathcal D_0\).

5.4. As in the proof of Lemma 3.1, \(\int_{\mathbb R}\#S(F,t)\,dt=\sum_e|dF(e)|\); if this is finite, \(\#S(F,t)<\infty\) for almost every \(t\). On \(\mathbb Z\) let \(F\) vanish on the negative integers and consist, on the positive integers, of excursions placed one after another: the \(j\)th excursion, \(j\ge1\), goes from \(0\) up to \(1\) in \(m=4^j\) steps of \(\frac1m\) and back down to \(0\) in \(m\) steps of \(\frac1m\). Its energy is \(2m\cdot m^{-2}=2\cdot4^{-j}\), so \(\mathcal E(F)<\infty\). A regular level \(t\in(0,1)\) is not a value of \(F\), so on the way up and on the way down of every excursion it lies strictly between two consecutive values; it is crossed twice by every excursion, and the cut is infinite.

5.5. Closedness of \(\mathcal D\) makes it a Hilbert space, which gives the representer \(r_b\) and the minimizer in Steps 1 and 2. Regularity of the level gives the strict inequalities \(f(v)<t<f(u)\) in Step 5, hence a strict decrease of energy, and it makes every vertex lie in one of the two sides in Step 6. Lemma 1.1 enters twice: through Lemma 2.1, and in Step 6 to write the restricted cochain as a gradient.

References