Baumslag-Gersten Group Overview

• The Baumslag-Gersten group is a finitely presented one-relator group defined by the relation (bab⁻¹)a(bab⁻¹)⁻¹ = a² and structured as an HNN extension of BS(1,2).
• Its extreme large-scale behavior is captured by a tower-type Dehn function and distorted subgroups, posing challenges for conventional algorithmic approaches while enabling efficient solutions via power circuits.
• The group exhibits remarkable endomorphic rigidity and serves as a key counterexample in geometric, algorithmic, and representation-theoretic investigations within combinatorial group theory.

The Baumslag–Gersten group is a finitely presented one-relator group that occupies a canonical place among pathological examples in combinatorial and geometric group theory. In standard notation it is the group usually denoted $\mathrm{BG}(1,2)$, $G_{(1,2)}$, $G$, or $\mathcal B$, and it is most commonly presented as

$$\mathcal B=\langle a,b \mid (bab^{-1})a(bab^{-1})^{-1}=a^2\rangle.$$

Its significance comes from the combination of a very small presentation with extreme large-scale behavior: the group has a Dehn function of tower type, very distorted subgroups, and a long history as a test case for limits of geometric, algorithmic, and representation-theoretic methods (Nyberg-Brodda, 25 Jun 2026, Miasnikov et al., 2011).

1. Definition, notation, and HNN structure

A basic structural observation is obtained by introducing

$c=bab^{-1}.$

The defining relation becomes

$cac^{-1}=a^2,$

so the subgroup generated by $a$ and $c$ is the Baumslag–Solitar group

$\BS(1,2)=\langle a,c \mid cac^{-1}=a^2\rangle.$

The Baumslag–Gersten group is therefore an HNN extension of $G_{(1,2)}$0 with stable letter $G_{(1,2)}$1 conjugating $G_{(1,2)}$2 onto $G_{(1,2)}$3 (Nyberg-Brodda, 25 Jun 2026). This HNN description is the organizing framework behind Britton-reduction arguments, automorphism calculations, and many algorithmic treatments.

Equivalent presentations are used throughout the literature. One common form is

$G_{(1,2)}$4

which exhibits the group as an iterated HNN extension: first $G_{(1,2)}$5, then an outer HNN step with stable letter $G_{(1,2)}$6 sending $G_{(1,2)}$7 to $G_{(1,2)}$8 (Lishak, 2015). Another standard form is

$G_{(1,2)}$9

which is equivalent to the one-relator presentation above (Miasnikov et al., 2011).

The internal structure of $G$0 is especially useful. It admits the semidirect-product description

$G$1

with multiplication

$G$2

and with $G$3, $G$4 in one standard choice of generators (Nyberg-Brodda, 25 Jun 2026). This model makes explicit the dyadic arithmetic underlying many computations in the group.

A point of terminology requires care. Several papers call $G$5 simply the “Baumslag group,” while also noting that this name has been used for other groups in the literature; the classical pathological one-relator group discussed here is the Baumslag–Gersten group in that sense (Korchagin, 2017).

2. Geometric pathology and large-scale behavior

The Baumslag–Gersten group is a standard example of a one-relator group with extreme isoperimetric behavior. Gersten proved that its Dehn function has a non-elementary lower bound, and Platonov proved that this lower bound is sharp; equivalently,

$G$6

or, in the notation $G$7, $G$8,

$G$9

Thus the Dehn function grows faster than any fixed-height tower of exponentials (Miasnikov et al., 2011, Gillis, 29 Jul 2025).

This behavior places the group far outside the usual tame classes. The same sources record that the group is not hyperbolic, not automatic, and not even asynchronously automatic, precisely because its Dehn function is non-elementary (Miasnikov et al., 2011). It is also a standard example of a one-relator group with very distorted subgroups and “exotic behavior” in combinatorial group theory (Nyberg-Brodda, 25 Jun 2026).

Residual properties are equally pathological. Baumslag showed that all finite quotients are cyclic, so the group is not residually finite; consequently it is not linear (Miasnikov et al., 2011). These features help explain why the group has repeatedly served as a boundary case in attempts to relate geometric, algebraic, and algorithmic complexity.

A common misconception is that such an extreme Dehn function should force the word problem to be comparably hard. The subsequent algorithmic theory shows that this implication fails in a particularly strong way for $\mathcal B$0; the geometric complexity measured by the Dehn function and the decision complexity of the word problem separate sharply in this example (Miasnikov et al., 2011).

3. Word problem, power circuits, and complexity theory

The classical Magnus breakdown procedure and iterative Britton reduction solve the word problem in principle, but on this group they generate exponents of non-elementary size. Myasnikov, Ushakov, and Won overcame this obstruction by introducing power circuits, a compressed representation of integers supporting addition and the operation $\mathcal B$1, and used them to prove that the word problem in $\mathcal B$2 is decidable in polynomial time, with running time $\mathcal B$3 in the original implementation (Miasnikov et al., 2011).

Power circuits are DAG-based data structures capable of representing integers whose ordinary binary length is not bounded by any fixed tower of exponentials in the size of the circuit. In the Baumslag–Gersten setting, they are matched exactly to the HNN relations

$\mathcal B$4

and to the exponent growth created by repeated Britton reductions (Miasnikov et al., 2011). Subsequent work refined both the data structure and the group-theoretic reduction scheme. The sequential bound was improved from $\mathcal B$5 to $\mathcal B$6, the word problem was shown to lie in $\mathcal B$7, and then in DLOGTIME-uniform $\mathcal B$8 for the broader family $\mathcal B$9, hence in particular for the Baumslag–Gersten group $$\mathcal B=\langle a,b \mid (bab^{-1})a(bab^{-1})^{-1}=a^2\rangle.$$0 (Mattes et al., 2021, Mattes et al., 2022).

This line of work makes the group a central counterexample to any naive identification of Dehn function growth with computational hardness. The Dehn function is of tower type, but the word problem is highly parallelizable. The explanation is structural: the algorithms do not build van Kampen diagrams; they manipulate enormous exponents directly in compressed form, using the HNN decomposition and arithmetic on power circuits (Mattes et al., 2021).

The conjugacy problem is more delicate. In $$\mathcal B=\langle a,b \mid (bab^{-1})a(bab^{-1})^{-1}=a^2\rangle.$$1 it is $$\mathcal B=\langle a,b \mid (bab^{-1})a(bab^{-1})^{-1}=a^2\rangle.$$2-complete, but in the Baumslag–Gersten group the HNN extension over $$\mathcal B=\langle a,b \mid (bab^{-1})a(bab^{-1})^{-1}=a^2\rangle.$$3 introduces substantially harder behavior (Diekert et al., 2013). Even so, conjugacy is decidable, and there is a strongly generic polynomial-time algorithm: outside an exponentially negligible set of inputs, conjugacy in $$\mathcal B=\langle a,b \mid (bab^{-1})a(bab^{-1})^{-1}=a^2\rangle.$$4 can be solved in $$\mathcal B=\langle a,b \mid (bab^{-1})a(bab^{-1})^{-1}=a^2\rangle.$$5 time (Diekert et al., 2013). Later work strengthened this to a strongly generic $$\mathcal B=\langle a,b \mid (bab^{-1})a(bab^{-1})^{-1}=a^2\rangle.$$6 bound for the family $$\mathcal B=\langle a,b \mid (bab^{-1})a(bab^{-1})^{-1}=a^2\rangle.$$7, and showed that for every fixed $$\mathcal B=\langle a,b \mid (bab^{-1})a(bab^{-1})^{-1}=a^2\rangle.$$8, hence for every fixed $$\mathcal B=\langle a,b \mid (bab^{-1})a(bab^{-1})^{-1}=a^2\rangle.$$9, the problem “is $c=bab^{-1}.$0 conjugate to $c=bab^{-1}.$1?” lies in $c=bab^{-1}.$2 on all inputs (Mattes et al., 2022).

4. Endomorphisms, rigidity, and outer automorphisms

Homomorphisms from the Baumslag–Gersten group to itself are unusually rigid. A complete classification shows that if

$c=bab^{-1}.$3

then every endomorphism $c=bab^{-1}.$4 either kills $c=bab^{-1}.$5 and $c=bab^{-1}.$6, or is an automorphism which, after composition with an inner automorphism, has the form

$c=bab^{-1}.$7

for some $c=bab^{-1}.$8, where $c=bab^{-1}.$9 and $cac^{-1}=a^2,$0 is the centralizer of $cac^{-1}=a^2,$1 in $cac^{-1}=a^2,$2 (Lishak, 2015).

From this description it follows that the Baumslag–Gersten group is both Hopfian and co-Hopfian: every surjective endomorphism is an automorphism, and every injective endomorphism is an automorphism (Lishak, 2015). This is striking because many Baumslag–Solitar-type groups display opposite behavior, whereas $cac^{-1}=a^2,$3 combines geometric pathology with strong endomorphic rigidity.

The same analysis computes the outer automorphism group: $cac^{-1}=a^2,$4 the dyadic rationals under addition (Lishak, 2015). Concretely, $cac^{-1}=a^2,$5 consists of the elements $cac^{-1}=a^2,$6, and the class of the automorphism

$cac^{-1}=a^2,$7

corresponds to the dyadic rational $cac^{-1}=a^2,$8. In particular, $cac^{-1}=a^2,$9 is not finitely generated (Lishak, 2015).

These results were already obtained by Andrew M. Brunner; the cited paper gives a self-contained exposition in the specific case of the Baumslag–Gersten group (Lishak, 2015).

5. Representations, germs of functions, and MF-approximations

A representation-theoretic question posed by A. Yu. Olshanskii asked whether the assignment

$a$0

embeds the Baumslag–Gersten group into the group $a$1 of germs at $a$2 of monotonically increasing-to-$a$3 continuous functions $a$4, with composition as the group law (Nyberg-Brodda, 25 Jun 2026). The defining relation is satisfied at the level of germs, so one obtains a quotient map

$a$5

A 2026 result shows that this representation is not faithful, giving a negative solution to Olshanskii’s Problem 17.99 from the Kourovka Notebook (Nyberg-Brodda, 25 Jun 2026). The proof defines

$a$6

computes their images as germs, and observes that $a$7 acts by $a$8, so $a$9 and $c$0 commute as germs. Hence the commutator $c$1 lies in the kernel. Britton’s Lemma, together with the description $c$2, shows that $c$3 in $c$4 (Nyberg-Brodda, 25 Jun 2026). The same paper notes that the finitely generated germ group $c$5 is still poorly understood: its abelianization is infinite cyclic, and whether it is finitely presented is left open (Nyberg-Brodda, 25 Jun 2026).

A rather different approximation-theoretic result goes in the opposite direction. The Baumslag–Gersten group has the MF-property: it embeds into the unitary group of

$c$6

equivalently it admits a faithful asymptotic matrix representation in operator norm (Korchagin, 2017). The proof rewrites the group as

$c$7

where $c$8 is a bi-infinite chain of Baumslag–Solitar-type pieces and $c$9 shifts the indices, then constructs carefully engineered finite-dimensional almost-representations of finite subgroups $\BS(1,2)=\langle a,c \mid cac^{-1}=a^2\rangle.$0 with controlled spectra (Korchagin, 2017). The result is notable because the group is not residually finite and has pathological finite-dimensional representation theory, yet still admits strong operator-norm approximations. The same paper emphasizes that the group was already known to be sofic and hyperlinear; MF is a stronger approximation property in this setting (Korchagin, 2017).

6. Conjugator length, related group classes, and current position

The conjugator length function of a finitely presented group measures how long a shortest conjugator may need to be, as a function of the lengths of two conjugate words. For the Baumslag–Gersten group, recent work proves upper and lower bounds of tower type: $\BS(1,2)=\langle a,c \mid cac^{-1}=a^2\rangle.$1 In particular, the conjugator length grows faster than any tower of exponentials of fixed height (Gillis, 29 Jul 2025). This mirrors the extremal behavior of the Dehn function and quantifies, in a separate way, how far conjugacy in the group can be from the behavior seen in ordinary Baumslag–Solitar groups.

The same paper places the group inside a graded family of iterated Baumslag–Solitar groups

$\BS(1,2)=\langle a,c \mid cac^{-1}=a^2\rangle.$2

for which

$\BS(1,2)=\langle a,c \mid cac^{-1}=a^2\rangle.$3

The Baumslag–Gersten group contains copies of these groups at increasing scales, and this is the mechanism behind the logarithmic-height tower bounds (Gillis, 29 Jul 2025). The authors conjecture that no one-relator group has a larger conjugator length function than the Baumslag–Gersten group (Gillis, 29 Jul 2025).

Comparison with nearby Bass–Serre classes is instructive. Higher-rank generalized Baumslag–Solitar groups admit a classification by residual finiteness and separability, but the Baumslag–Gersten group is not itself a generalized Baumslag–Solitar group, so those classification results do not apply directly to it (Zearra et al., 2023). This underscores a broader point: although $\BS(1,2)=\langle a,c \mid cac^{-1}=a^2\rangle.$4 is assembled from Baumslag–Solitar pieces and is governed at many stages by HNN technology, it lies beyond the more rigid graph-of-$\BS(1,2)=\langle a,c \mid cac^{-1}=a^2\rangle.$5-groups frameworks where separability behavior is controlled.

The resulting picture is unusually coherent. The Baumslag–Gersten group is a one-relator HNN extension of $\BS(1,2)=\langle a,c \mid cac^{-1}=a^2\rangle.$6; it is non-residually finite, non-linear, and geometrically extreme, with a Dehn function and conjugator length of tower type (Miasnikov et al., 2011, Gillis, 29 Jul 2025). At the same time, it is algorithmically tractable for the word problem through power-circuit compression, rigid enough to be Hopfian and co-Hopfian, subtle enough to defeat a natural germ representation by $\BS(1,2)=\langle a,c \mid cac^{-1}=a^2\rangle.$7 and $\BS(1,2)=\langle a,c \mid cac^{-1}=a^2\rangle.$8, and flexible enough to admit faithful MF-approximations by matrices (Lishak, 2015, Nyberg-Brodda, 25 Jun 2026, Korchagin, 2017). Few finitely presented groups concentrate so many boundary phenomena in so small a presentation.