Effective Følner Sequences

• Effective Følner sequences are explicitly computable witnesses of group amenability that enable algorithms to produce n-Følner sets validating approximate invariance.
• They demonstrate that computable Følner sets can exist even in groups with unsolvable word problems and under complex group extensions with controlled distortion.
• Explicit constructions in ergodic theory, operator algebras, and topological groups highlight the broad impact and practical applications of effective Følner methods.

Effective Følner sequences are algorithmically or explicitly realizable witnesses of amenability. In the finitely generated setting, the basic formulation is usually the computability of Følner sets: given a parameter $n$, one must effectively produce a finite set whose image in the group is $n$-invariant in the Følner sense. Subsequent work broadened this perspective to computable Reiter functions, subrecursive Følner functions, computable Følner monotilings, effective subsequences used in ergodic theorems, and several operator-algebraic and coarse-geometric analogues (Cavaleri, 2016, Cavaleri, 2017, Moriakov, 2015, Chakraborty et al., 15 Dec 2025).

1. Foundational definitions

For a finitely generated group $\Gamma = \langle X \rangle$, an $n$-Følner set is a non-empty finite subset $F \subset \Gamma$ such that

$$\forall x \in X: \frac{|F \setminus xF|}{|F|} < n^{-1}.$$

The group $\Gamma$ is amenable if it admits such sets for every $n \in \mathbb{N}$ (Cavaleri, 2016).

The effective version studied by Cavaleri is formulated in the free group over the generators. The group $\Gamma$ has computable Følner sets with respect to $X$ if there is an algorithm that, given $n$0, outputs a finite subset $n$1 so that the image of $n$2 in $n$3 is an $n$4-Følner set. The associated Følner function is

$n$5

and this notion of computability is independent of the generating set (Cavaleri, 2016).

A related formulation, used in the study of effective amenability for recursively presented groups, replaces $n$6 by a symmetric-difference condition: $n$7 That paper also isolates several algorithmic variants: computable Følner sets, computable Følner sets by one-to-one preimages, computable Reiter functions, and subrecursive Følner functions (Cavaleri, 2017).

The basic conceptual distinction is between existence and production. Classical amenability asks only that Følner witnesses exist. Effective Følner theory asks whether they can be produced, uniformly in the approximation parameter, by algorithms or explicit constructions.

2. Computability without solvable word problem

A decisive example is provided by the Kharlampovich group $n$8, a finitely presented solvable group with unsolvable word problem. Cavaleri gives an explicit description and an algorithm to compute Følner sets for these groups, proving that computable Følner sets can exist even if the word problem is unsolvable (Cavaleri, 2016).

The construction uses a decomposition of $n$9 involving Abelian groups and describes Følner sets in the form

$\Gamma = \langle X \rangle$0

with generalized cube sets built from commutative subgroups. For $\Gamma = \langle X \rangle$1, where $\Gamma = \langle X \rangle$2 is the exponent appearing in the construction, the description stabilizes and becomes effective. Cavaleri also records the upper bound

$\Gamma = \langle X \rangle$3

for the Følner function in this case (Cavaleri, 2016).

This example changes the logical status of effective amenability. It shows that computable Følner sets do not characterize solvability of the word problem. In particular, the class of finitely presented groups with computable Følner sets is strictly larger than the class of finitely presented groups with solvable word problem, answering Vershik’s question in the positive (Cavaleri, 2016).

A common misconception is therefore that effective Følner constructions require a decidable word problem. The Kharlampovich example shows that this is false for computable Følner sets themselves, even though stronger injectivity requirements on preimages do recover the word problem, as discussed below.

3. Stability under extensions and distortion

A substantial part of the theory concerns permanence under extensions. Cavaleri proves several closure results and gives explicit upper bounds for the Følner function in each case (Cavaleri, 2016).

In Abelian extensions

$\Gamma = \langle X \rangle$4

if $\Gamma = \langle X \rangle$5 is Abelian, $\Gamma = \langle X \rangle$6 is amenable with solvable word problem, and $\Gamma = \langle X \rangle$7 is finitely presented, then $\Gamma = \langle X \rangle$8 has computable Følner sets. The corresponding upper bound is

$\Gamma = \langle X \rangle$9

For semidirect products, if $n$0 and $n$1 have computable Følner sets, then their semidirect product does too. In that situation one has

$n$2

where $n$3 and $n$4 depend on the generating sets and the action.

In the most general extension theorem, for

$n$5

if $n$6 has computable Følner sets, $n$7 is amenable with solvable word problem, and the distortion function

$n$8

is subrecursive, then $n$9 has computable Følner sets. The accompanying upper bound is

$F \subset \Gamma$0

Cavaleri emphasizes that subrecursivity of both Følner and distortion functions is needed on the algorithmic side (Cavaleri, 2016).

These results place effective Følner theory within the standard extension calculus of geometric group theory. They also clarify that the main obstruction is not amenability of the quotient alone, but the ability to control the cost of lifting approximate invariance through the kernel.

4. Effective amenability, decision problems, and complexity

Cavaleri’s later paper organizes several notions of effective amenability for recursively presented amenable groups (Cavaleri, 2017).

Notion	Requirement	Relationship
Computable Følner sets (CCF)	algorithm outputs finite $F \subset \Gamma$1 whose image is $F \subset \Gamma$2-Følner	$F \subset \Gamma$3
Computable Reiter functions (CCR)	algorithm outputs finitely supported $F \subset \Gamma$4 with $F \subset \Gamma$5-invariant pushforward	$F \subset \Gamma$6
Subrecursive Følner function (CSF)	$F \subset \Gamma$7 is bounded above by a recursive function	holds for all recursively presented amenable groups

The main structural theorem states that recursively presented amenable groups have subrecursive Følner function, answering a question of Gromov. More precisely, every recursively enumerable class of recursive amenable presentations admits a uniform recursive upper bound for the asymptotic growth of the corresponding Følner functions (Cavaleri, 2017).

At the same time, the paper proves

$F \subset \Gamma$8

Thus, for recursively presented amenable groups, computable Følner sets by injective preimages are equivalent to solvability of the Word Problem, while computable Reiter functions and subrecursive Følner functions exist throughout the amenable class (Cavaleri, 2017).

The same work shows that, for recursively presented amenable groups, solvability of the Equality Problem on a generic set is equivalent to solvability of the Word Problem on the whole group. In particular, finitely presented amenable groups can have unsolvable generic Equality Problem, and the Kharlampovich groups provide such examples (Cavaleri, 2017).

A further extension removes finite generation. For a computably enumerable group $F \subset \Gamma$9, possibly not finitely generated, the following are equivalent: amenability, computable Reiter functions, subrecursive Følner function, and $$\forall x \in X: \frac{|F \setminus xF|}{|F|} < n^{-1}.$$0-amenability. If the group is computable, then computable amenability is equivalent to computability of the group. In the same paper, the class of indices coding effective Følner sequences for a computable group is shown to belong to $$\forall x \in X: \frac{|F \setminus xF|}{|F|} < n^{-1}.$$1, and in certain abelian groups this classification is sharp: the class is $$\forall x \in X: \frac{|F \setminus xF|}{|F|} < n^{-1}.$$2-complete (Duda et al., 15 Sep 2025).

Several basic questions remain open. Cavaleri asks whether all finitely generated solvable groups have computable Følner sets, whether computability of Følner sets is stable under quotients, and whether a subrecursive Følner function implies computability of Følner sets. The paper notes that a positive answer to the third question would imply positive answers to the first two (Cavaleri, 2016).

5. Explicit constructions, exact sets, and monotilings

Effective Følner theory is not restricted to abstract existence theorems. In several families the relevant sets can be written down explicitly.

In the context of computable dynamics, Moriakov introduces computable Følner monotilings $$\forall x \in X: \frac{|F \setminus xF|}{|F|} < n^{-1}.$$3, where $$\forall x \in X: \frac{|F \setminus xF|}{|F|} < n^{-1}.$$4 is a finite tile, $$\forall x \in X: \frac{|F \setminus xF|}{|F|} < n^{-1}.$$5 is a set of centers, the translates $$\forall x \in X: \frac{|F \setminus xF|}{|F|} < n^{-1}.$$6 tile the group, and $$\forall x \in X: \frac{|F \setminus xF|}{|F|} < n^{-1}.$$7 is a Følner sequence. A left computable Følner monotiling requires $$\forall x \in X: \frac{|F \setminus xF|}{|F|} < n^{-1}.$$8 to be canonically computable and $$\forall x \in X: \frac{|F \setminus xF|}{|F|} < n^{-1}.$$9 to be computable. Regular monotilings add tempered two-sidedness, a good weight condition for $\Gamma$0, growth $\Gamma$1, and the requirement $\Gamma$2 for all $\Gamma$3 (Moriakov, 2015).

For $\Gamma$4, the paper gives the explicit choice

$\Gamma$5

For the discrete Heisenberg group $\Gamma$6, it gives

$\Gamma$7

$\Gamma$8

The paper states that for every $\Gamma$9, $n \in \mathbb{N}$0 and the groups of unipotent upper-triangular matrices of dimension $n \in \mathbb{N}$1 with integer entries admit computable regular symmetric Følner monotilings, and that the required computing algorithms can be provided explicitly (Moriakov, 2015).

A different explicitness result concerns exact Følner functions. For any finite group $n \in \mathbb{N}$2 and all $n \in \mathbb{N}$3, the wreath product $n \in \mathbb{N}$4 satisfies

$n \in \mathbb{N}$5

and the key Følner sets are

$n \in \mathbb{N}$6

For $n \in \mathbb{N}$7, the standard sets

$n \in \mathbb{N}$8

minimize the edge-boundary ratio among sets of size up to $n \in \mathbb{N}$9 (Stankov, 2021).

These examples show that “effective” can mean more than recursive enumerability. In some classes, the Følner geometry is explicit enough to permit exact formulas, optimality proofs, and canonical tilings.

6. Broader variants and analogues

In ergodic theory, an effective Følner sequence may mean one that supports a quantitative reduction to $\Gamma$0-actions. For every two-sided Følner sequence $\Gamma$1 in a unimodular amenable group, there exists a subsequence $\Gamma$2, a constant $\Gamma$3, a strictly increasing function $\Gamma$4, and a positive linear Markov operator $\Gamma$5 such that, for all positive $\Gamma$6 in $\Gamma$7, $\Gamma$8, and sufficiently large $\Gamma$9,

$X$0

The paper stresses that every two-sided Følner sequence has a subsequence satisfying the technical conditions needed for this dominance argument (Chakraborty et al., 15 Dec 2025).

For countable amenable groups, increasing Tempelman Følner sequences admit a combinatorial tiling property for pmp actions, and this directly implies the pointwise ergodic theorem. The Tempelman condition

$X$1

provides the control needed for iterated Vitali covering arguments (Boretsky et al., 2019).

Beyond discrete groups, Schneider and Thom extend Følner’s amenability criterion to topological groups by replacing overlap counts with matching numbers. For finite $X$2 and an identity neighborhood $X$3, amenability is characterized by the existence of finite $X$4 such that

$X$5

for prescribed $X$6, $X$7, and $X$8 (Schneider et al., 2016).

In coarse geometry, the analogue of effective Følner data appears in Yu’s Property A. For discrete bounded geometry spaces which coarsely have unbounded components, for all countable discrete groups, and for all box spaces, Property A is equivalent to naive Property A, meaning that the generalized Følner sets can be chosen as actual subsets of the space rather than weighted subsets of $X$9 (Niblo et al., 2024).

Operator theory provides another analogue. A sequence of nonzero finite-rank orthogonal projections $n$00 is a Følner sequence for a set of operators $n$01 if

$n$02

Any essentially normal operator has a proper Følner sequence, and amenable traces on a unital separable C$n$03-algebra can be approximated by the states

$n$04

associated to such sequences (Ara et al., 2013, Ara et al., 2012).

A final direction is deliberately asymmetric Følner geometry. The notion of a left scheme requires finite sets $n$05 with

$n$06

thereby combining summable left boundaries with displacement under right translation. This mechanism produces left/right asymmetry in $n$07-Dirichlet spaces for non-virtually abelian finitely generated nilpotent groups and also applies to amenable wreath products over $n$08 and solvable Baumslag–Solitar groups (Avraham-Re'em et al., 12 May 2026).

Taken together, these developments show that effective Følner sequences are not a single construction but a family of closely related formalisms. In the group-theoretic core, they quantify the algorithmic content of amenability. In adjacent areas, they supply explicit windows for entropy and ergodic limits, matching criteria in topological groups, subset-valued witnesses for Property A, finite-rank approximants in operator algebras, and even asymmetric geometric mechanisms that go beyond the classical balanced Følner paradigm.