The same group can have actions with different costs

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Family 259 · Group theory
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-22

1. The question

Suppose a countable group acts on a probability space. Each point has an orbit, consisting of all points reachable by applying group elements. How many edges, on average, are needed to connect every orbit?

The answer is called the cost of the action. One may use different group moves on different measurable subsets, so the minimum can be much smaller than the number of generators in a presentation. The fixed-price problem asks whether all essentially free probability-preserving actions of the same group nevertheless have the same cost.

The manuscript claims a negative answer. It gives one finitely generated group with a Bernoulli action whose cost stays strictly above one, while actions obtained by adding a finite height coordinate have costs approaching one. The group is unchanged; the measurable organization of its orbits changes the minimum cost.

2. Background and essential definitions

Actions and orbit relations

A probability-measure-preserving action of a countable group $\Gamma$ on $(X,\mu)$ assigns a measurable, measure-preserving bijection to each group element. The paper uses right actions, so $(xg)h=x(gh)$.

The orbit relation is

$$R_{\Gamma\curvearrowright X} =\{(x,xg):x\in X,\ g\in\Gamma\}.$$

It records which points are connected by the action, without recording which particular group element makes the connection.

An action is essentially free if every nonidentity element fixes a set of measure zero. After discarding one invariant null set, every remaining point has trivial stabilizer. This condition prevents stabilizers from artificially reducing the complexity of orbits.

Graphings and cost

A graphing is a countable collection of partial measure-preserving bijections

$$\varphi_i:D_i\longrightarrow E_i$$

whose graphs lie inside the orbit relation. These are allowable edges between points in the same orbit. The graphing generates the relation if finite paths using the maps and their inverses connect all orbit-related points outside a null set.

Its cost is

$$C(\{\varphi_i\})=\sum_i\mu(D_i),$$

and the cost of the relation is the infimum over generating graphings. An inverse edge is available for traversing a path; it is not separately charged when it is the inverse of a listed map.

Using every group generator on all of $X$ gives an immediate upper bound. Better graphings can restrict expensive moves to small sets and reach those sets through other available moves. Cost is therefore a measurable optimization problem, not simply the size of a generating set.

A group has fixed price if this infimum is the same for all its essentially free probability-preserving actions. Free groups of rank $r$ have fixed price $r$, while many groups with commuting or amenable structure have fixed price one.

Bernoulli actions and height extensions

The Bernoulli space $[0,1]^\Gamma$ assigns an independent uniform random label to every group element. The group shifts the coordinate labels. Its action is essentially free; the paper restricts to the invariant set $X$ of points with trivial stabilizer, which has full measure.

A finite height extension adds a uniformly distributed coordinate in $\mathbb Z/M\mathbb Z$. Group elements move both the Bernoulli configuration and the height. The finite coordinate does not introduce stabilizers because the Bernoulli coordinate already makes the action free.

3. Development history and previous partial results

Levitt introduced cost as an invariant of measured equivalence relations in the 1990s. Gaboriau developed its connection with group theory, including fixed-price results for free groups and a framework for calculating costs of relations assembled from simpler pieces.

These results showed that the invariant can detect substantial algebraic structure. A free group needs a definite average number of orbit edges, whereas commuting directions can let one restrict many moves to small complete sets and reuse them elsewhere. A complete set here means one meeting every orbit of the relation under consideration.

Many positive fixed-price theorems accumulated. The manuscript cites, among recent developments, Khezeli’s theorem that products of two infinite countable groups have fixed price one, Bevilacqua–Bowen’s metric criteria, and Slutsky’s criterion using small product neighborhoods of finite subsets. These results cover important classes without proving that every countable group has fixed price.

The new example deliberately combines two contrasting structures. A free factor supplies many independent directions, while a direct-product factor makes some directions cheap. They are glued along a common subgroup in a way that lets an extra finite height coordinate reduce the cost further.

The lower-bound method has another ancestry: Arzhantseva–Ol’shanskii developed graph-shortening arguments that force large generator rank in suitable groups. Kapovich and Schupp refined this approach. The manuscript connects such finite presentation estimates to graphings of a Bernoulli action, rather than relying only on an abstract cost formula for the amalgam.

4. What the new manuscripts claim

The October 5, 2026 manuscript specifies

$$\begin{aligned} A&=F(a,b_1,\ldots,b_{99}),\\ w&=ab_1ab_2\cdots ab_{99}a,\\ J&=\langle b_1,\ldots,b_{99},w\rangle\leq A. \end{aligned}$$

Thus $A$ is free of rank 100. It then forms

$$\Gamma=A*_J\bigl(J\times\langle t\rangle\bigr).$$

An amalgamated free product glues two groups by identifying the displayed copies of $J$. In this example, it can be viewed as adjoining $t$ to $A$ and requiring $t$ to commute with every element of $J$.

There is a height homomorphism $\chi:\Gamma\to\mathbb Z$ with

$$\chi(a)=1,\qquad \chi(b_i)=\chi(t)=0.$$

In particular, $\chi(w)=100$. The height is not zero on all of $J$.

Let $X$ carry the free Bernoulli action. On $Y_M=X\times\mathbb Z/M\mathbb Z$, use the action

$$(x,r)g=(xg,r+\chi(g)\bmod M).$$

The theorem claims a fixed constant $\eta>0$, independent of $M$, such that

$$\begin{aligned} \operatorname{Cost}(R_{\Gamma\curvearrowright X})&\geq1+\eta,\\ 1\leq\operatorname{Cost}(R_{\Gamma\curvearrowright Y_M}) &\leq1+\frac{99}{M}\qquad(M>100). \end{aligned}$$

The paper gives explicit conditions defining a sufficiently small positive $\eta$. Choosing $M>99/\eta$, as well as $M>100$, separates the two costs strictly. Exact values are unnecessary for disproving fixed price.

5. Why the advance matters

Importance: 5/5, provisional. If correct, the example disproves the general fixed-price assertion, a central question about how much orbit complexity is determined by a group alone.

The phenomenon is particularly clear because the low-cost actions are finite extensions of a familiar Bernoulli action. A finite additional coordinate can organize orbit connections more economically, even though the action remains free and the acting group remains the same.

The theorem does not diminish the earlier fixed-price results for their respective classes. Instead, it identifies a limit to generalizing them. The group is an amalgam involving a product, not itself merely a product of two infinite groups to which a product fixed-price theorem would apply directly.

The argument also offers a bridge between measurable and finite mathematics: a lower bound for an infimum over all measurable graphings is extracted from generator-rank bounds in finite permutation models.

6. The main idea

Why the extra height coordinate makes edges cheap

Write $B=J\times\langle t\rangle$. Keep all $t$-edges, at cost one. Choose a small measurable set of measure $p$ meeting every $t$-orbit and retain each of the 100 displayed generators of $J$ only on that set.

Because $t$ commutes with $J$, a point can travel along its $t$-orbit to the small set, apply a $J$-generator there, and travel back. These edges generate the $B$-relation at cost $1+100p$, arbitrarily close to one.

It remains to supply $a$-edges. In the height extension, keep them only at 99 consecutive heights, costing $99/M$. The identity

$$w=ab_1ab_2\cdots ab_{99}a$$

propagates availability to the next height. Starting at the first height, the 99 known $a$-moves, interspersed with available $b_i$-moves, reach the last missing $a$-edge. The available $w$-move reaches its endpoint by another route, so that edge is already generated.

Repeating around the finite height cycle supplies every $a$-edge. Letting $p$ tend to zero gives the upper bound $1+99/M$.

Measuring what remains after the common subgroup is free

For the Bernoulli action, let $R_A$ and $R_J$ denote the relations from the subgroups. Define a relative cost

$$\delta_X=\inf\{C(\mathcal E): R_J\vee\mathcal E=R_A,\ \mathcal E\subseteq R_A\}.$$

The join symbol means “generate a relation using both kinds of steps.” Thus $\delta_X$ measures how much must still be paid to generate $A$-orbits when all $J$-steps are supplied without charge.

Adapting Gaboriau’s deployment and compression methods, the manuscript claims

$$\operatorname{Cost}(R_{\Gamma\curvearrowright X}) \geq1+\delta_X.$$

The construction subdivides graphing moves into factor moves, enlarges the supplied relation and keeps exact track of the cost saved by compression. This relative estimate is proved for the setting at hand; simply subtracting known costs of the factor groups would not give the needed conclusion.

Finite models force positive relative cost

Take a large finite set $V$ with a right action of the free group $A$ by permutations. Set $u_0=1$ and $u_i=u_{i-1}ab_i$. The finite model produces a group

$$D_V=\left\langle x_v\ (v\in V)\ \middle|\ x_vx_{vu_1}\cdots x_{vu_{99}}=1\ (v\in V) \right\rangle.$$

The symbols record $a$-steps after $J$-steps have been made free. Let $\operatorname{rk}(D_V)$ be the least number of generators of this group.

Along finite models in which each fixed nonidentity element of $A$ fixes a vanishing fraction of points, the manuscript transfers Bernoulli graphings to the inequality

$$\delta_X\geq \limsup\frac{\operatorname{rk}(D_V)}{|V|}.$$

These are finite permutation models of $A$, not asserted finite quotient approximations of the entire amalgam $\Gamma$.

Random permutations give models whose length-100 relation rows expand across many symbols and have few overlaps. The remaining task is a deterministic rank bound: a generating set that was too small could be represented by a minimal labelled graph, and the relations would force a long segment in that graph to admit a shorter replacement. This contradicts minimality without increasing graph rank.

Exceptional symbols are absorbed into their actual generated subgroup before shortening. All true relations of that coefficient subgroup are retained, even if it is not finitely presented. A planar word lemma controls the remaining letters while preserving those coefficient values. Expansion prevents the exceptional set from absorbing everything.

The result is a positive proportional rank bound, $\operatorname{rk}(D_V)\geq\eta|V|$ asymptotically. It forces $\delta_X\geq\eta$ and hence the Bernoulli cost lower bound.

7. Scope, limitations, and open questions

The conclusion compares essentially free probability-measure-preserving actions on standard probability spaces. Removing freeness or changing the measure setting would define a different problem.

The displayed bounds show that the finite height-extension costs converge to one. They do not identify the exact cost for each finite $M$, nor do they assert that one particular finite height extension has cost exactly one. The Bernoulli action’s exact cost is likewise not determined by its lower bound.

The constants are chosen to make an expansion estimate work, not to optimize the gap or the number of generators. The construction settles existence as claimed; it does not classify all groups without fixed price.

The example also does not say that every finite extension of an action lowers cost. Its special relation and height homomorphism provide the propagation mechanism. Understanding which other groups and actions admit comparable savings is a separate question from this counterexample.

8. Sources and further reading

The historical foundations are Levitt’s On the cost of generating an equivalence relation (1995) and Gaboriau’s Coût des relations d’équivalence et des groupes (2000), both cited in the manuscript. For recent positive results, see Khezeli’s product theorem, Bevilacqua–Bowen’s metric criteria, and Slutsky’s product-neighborhood criterion.

Kapovich–Schupp’s account of the Arzhantseva–Ol’shanskii method gives background for the finite rank obstruction. The source manuscript below supplies the explicit group, the two actions, and the relative-cost transfer used in this construction.

Source manuscripts

• M1. OpenAI, A group without fixed price (October 5, 2026).

Sources consulted on 22 October 2026 at repository revision 1581d4c. Download the manuscript citations.