---
title: Baumslag–Solitar Group
url: https://www.emergentmind.com/topics/baumslag-solitar-group
type: topic
---

# Baumslag–Solitar Group

The Baumslag–Solitar groups are a class of two-generator, one-relator groups introduced as fundamental examples in combinatorial and geometric group theory. Denoted $\mathrm{BS}(m,n)$ for nonzero integers $m$ and $n$, these groups are defined by the presentation
\[
\mathrm{BS}(m,n) = \langle a, t \mid t\,a^m\,t^{-1} = a^n \rangle,
\]
realizing them as HNN-extensions of the infinite cyclic group $\langle a \rangle$ mapping $a^m \mapsto a^n$. They serve as prototypical cases in the study of non-Hopfian groups, groups with exotic algorithmic, residual, and isoperimetric properties, and as building blocks for the broader class of generalized Baumslag–Solitar (GBS) groups.

## 1. Structural Foundations and Group Theoretic Presentation

A Baumslag–Solitar group $\mathrm{BS}(m,n)$ is characterized by the integer parameters $m$ and $n$, and their structure is determined by the underlying HNN-extension; that is, $\mathrm{BS}(m,n)$ is an amalgamation of the infinite cyclic group $\mathbb{Z}$ along the index-$m$ and index-$n$ subgroups associated to $a^m$ and $a^n$, so that the stable letter $t$ conjugates $a^m$ to $a^n$ [1111.3701][1602.02445]. Essential features arising from the values of $m$ and $n$ include:

- **Amenability and solvability:** $\mathrm{BS}(m,n)$ is amenable if and only if $|m|=1$ or $|n|=1$, with these cases reducing to virtually abelian groups. Specifically, $\mathrm{BS}(1,1)\cong\mathbb{Z}^2$ and $\mathrm{BS}(1,-1)\cong\mathbb{Z}\ast(\mathbb{Z}/2\mathbb{Z})$ [2405.09736].
- **Metabelian structure:** For $m=1$ or $n=1$ (excluding $(1,1),\ (1,-1)$), the group is metabelian; e.g., $\mathrm{BS}(1,n) \cong \mathbb{Z}[1/n] \rtimes \mathbb{Z}$, with multiplicative automorphism $t a t^{-1}=a^n$ [1003.0148].
- **Non-Hopfian property:** For certain non-unit and non-equal $(m, n)$, these groups provide canonical examples of finitely generated, non-Hopfian groups, i.e., admitting surjective but non-injective endomorphisms [1602.02445][1308.5122].
- **Non-amenability:** If $|m|, |n|>1$, the group is non-amenable.

Generalized Baumslag–Solitar (GBS) groups are defined as fundamental groups of finite graphs of groups with all vertex and edge groups infinite cyclic, obtaining $\mathrm{BS}(m,n)$ as the one-vertex, one-loop case [2405.09736][2406.03960][1602.02445].

## 2. Subgroup Structure, Embeddings, and Quotients

The subgroup and quotient landscape of Baumslag–Solitar groups displays rich arithmetic and combinatorial structure, thoroughly elucidated in recent work [1308.5122][2210.14990].

- **Embedding Criteria:** $\mathrm{BS}(r,s)$ embeds in $\mathrm{BS}(m,n)$ if and only if (i) $\frac{r}{s}$ is a rational power of $\frac{m}{n}$, (ii) all prime divisors of $rs$ divide $mn$, subject to a refined divisibility constraint at the level of exponents in the prime decompositions, and (iii) the unimodular ($\pm1$) exponents condition [1308.5122].
- **Quotients and epi-equivalence:** For fixed $(m,n)$, all non-elementary two-generated GBS-quotients of $\mathrm{BS}(m,n)$ are classified according to the topology and label structure of reduced labelled graphs (segment, circle, or lollipop types). $\mathrm{BS}(m,n)$ is non-Hopfian if $m,n$ differ in their set of prime divisors, resulting in infinitely many non-isomorphic quotients $G$ so that both $BS(m,n)\twoheadrightarrow G$ and $G\twoheadrightarrow BS(m,n)$ [1308.5122].
- **Subgroup Space Topology:** The Chabauty topology on the space of subgroups $\mathrm{Sub}(\Gamma)$, for $\Gamma=\mathrm{BS}(m,n)$, yields a perfect kernel $K$ comprising all infinite index subgroups (when $|m|,|n|>1$), with conjugacy dynamics that naturally partition $K$ according to an arithmetically defined "phenotype" invariant. On each phenotype stratum, the conjugation action is topologically transitive [2210.14990].

### Example Table: Subgroup Phenotype Partition for $\mathrm{BS}(m,n)$

| Phenotype $q$           | Openness/Closedness           | Conjugacy Dynamics         |
|-------------------------|-------------------------------|---------------------------|
| $q < \infty$            | Open (closed if $m = n$)      | Topologically transitive   |
| $q = \infty$            | Closed (not open if $m \ne n$)| Transitive action         |

In the Hopfian case ($m=n$), phenotype strata are both open and closed; for $m\ne n$, their closures accumulate on the $q=\infty$ piece.

## 3. Residual Properties, Cohomology, and Separability

Baumslag–Solitar and GBS groups are test cases for questions of residual finiteness, conjugacy separability, and cohomological "goodness" [2405.09736][2406.03960].

- **Residual Finiteness:** For $\mathrm{BS}(m,n)$, residual finiteness holds if and only if $\gcd(m, n) = 1$ or $|m|=|n|$; in non-coprime, non-equal cases, Meskin showed non-residual finiteness [2406.03960].
- **Conjugacy Separability:** The Sokolov theorems establish that for non-solvable GBS groups (including non-Hopfian $\mathrm{BS}(m,n)$), conjugacy finite-separability is equivalent to residual finiteness. For solvable cases ($\mathrm{BS}(1,n)$), one must further require the class of finite quotients detects all prime divisors [2405.09736].
- **Cohomological Separability ("goodness"):** The trichotomy for $\mathrm{BS}(m,n)$ [2406.03960]:

  1. $\gcd(m, n)=1$ or $|m|=|n|$: group is residually finite, $\mathrm{cd}=\mathrm{cd}(\widehat{\Gamma})=2$; group is cohomologically good.
  2. $m,n$ isocratic but not coprime, and $|m|\ne|n|$: not cohomologically good.
  3. Not isocratic: profinite completion acquires torsion, $\mathrm{cd}(\widehat{\Gamma}) = \infty$.

  For GBS groups, separable cohomology occurs only if every cycle in the labeled graph satisfies matching augmentation products.

## 4. Representation Theory and Zariski Closure

The finite-dimensional irreducible complex representations of coprime $\mathrm{BS}(p, q)$ are classified via metacyclic group structure and Zariski closure techniques [1112.3952]:

- **Existence and Structure:** An irreducible $(n+1)$-dimensional complex representation exists precisely for each divisor $\ell \mid q^{n+1} - p^{n+1}$, primitive $\ell$-th root of unity $\lambda$, and $s$ solving $p\equiv qs \pmod\ell$, with an additional non-divisibility ("no smaller divisor") condition.
- **Matrix Form:** Representations are conjugate to the form
  \[
  A = c\begin{pmatrix}
  0 & \ldots & 0 & 1 \\
  1 & 0 & \ldots & 0 \\
  & \ddots & \ddots & \vdots \\
  0 & \cdots & 1 & 0
  \end{pmatrix}, \quad
  B = \mathrm{diag}(\lambda, \lambda^s, \ldots, \lambda^{s^n}),
  \]
  where $c\in\mathbb{C}^\times$.
- **Classification:** The image of the representation lies in a metacyclic subgroup of $\mathrm{GL}_{n+1}(\mathbb{C})$, with normality and semisimplicity of the diagonal subgroup arising from Zariski considerations.

## 5. Algorithmic Complexity: Word and Conjugacy Problems

Algorithmic properties of $\mathrm{BS}(m,n)$ and GBS groups have been explicitly quantified [1602.02445]:

- **Word Problem:** For all GBS (and thus all Baumslag–Solitar) groups, the word problem is LOGSPACE-complete. The reduction constructs a colored Britton factorization and reduces to the free group word problem via uniform $\mathrm{TC}^0$ circuits.
- **Conjugacy Problem:** Also solvable in LOGSPACE (for fixed GBS groups), via reduction to arithmetic in $\mathbb{Z}$ and congruence systems; for the standard (non-uniform) version in $\mathrm{BS}(m,n)$ the reduction is $\mathrm{AC}^0$-Turing-equivalent to the free group word problem. The uniform conjugacy problem (where the graph-of-groups structure is part of the input) is EXPSPACE-complete.
- **Structural Insights:** The tractable complexity follows from the Bass–Serre theoretic normal forms, combinatorics of edge-paths and cancellation, and explicit manipulation of arithmetic invariants within Britton's strategy.

## 6. Metric, Geometric, and Combinatorial Properties

Baumslag–Solitar groups exhibit distinctive isoperimetric, geometric, and additive-combinatorial behaviors, particularly in the metabelian cases [1003.0148][2402.16128].

- **Dehn Functions:** For solvable cases ($\mathrm{BS}(1,n)$, $|n|\geq 2$), the Dehn function is exponential. However, each $\mathrm{BS}(1,n)$ embeds in an explicit finitely presented metabelian group with quadratic Dehn function $\Gamma_n$ [1003.0148].
- **Asymptotic Cones:** For $\Gamma_n$, the asymptotic cone is bilipschitz homeomorphic to a branched subset of a product of $\mathbb{R}$-trees, matching the Diestel–Leader structure of geometrically similar solvable groups.
- **Sumset Phenomena:** Additive combinatorics in $\mathrm{BS}(1,3)$ demonstrates sharp “small doubling” direct-inverse theorems. For $S = b a^A \subset \mathrm{BS}(1,3)$, $|S^2| = |A + 3A| \geq 4|A| - 4$, with classification of extremal cases and a Freiman-type structure theorem when $|S^2|$ is almost minimal [2402.16128]. These results generalize methods of Freiman–Herzog et al. for $BS(1,2)$, and plausible implications are that similar phenomena should extend to non-unimodular $\mathrm{BS}(m, n)$.

## 7. Ergodic Theory, Measure Equivalence, and Orbit Invariants

Actions of non-amenable $\mathrm{BS}(m,n)$ display invariants rigid under weak orbit equivalence and possess strong measure equivalence rigidity [1111.3701]:

- **Orbit Equivalence Invariants:** For ergodic, essentially free, probability measure-preserving actions, there is an associated $\mathbb{R}$-flow (arising from the modular homomorphism), which is invariant under weak orbit equivalence.
- **ME-Rigidity:** If $H$ has an infinite amenable normal subgroup and non-elementary Gromov-hyperbolic quotient, it is not measure equivalent to $BS(p,q)$ for $|p|<|q|$. Thus, distinct Baumslag–Solitar groups with distinct moduli are ME-distinguished.
- **Classification of Actions:** For certain ergodic subactions, WOE-rigidity can force the pair $(p, q)$ up to symmetries. However, there exist examples with WOE but non-conjugate actions not classified by the flow invariant.

---

**References**:  
[1111.3701]: Invariants of orbit equivalence relations and Baumslag-Solitar groups  
[1112.3952]: Irreducible Representations of Baumslag-Solitar Groups  
[1308.5122]: Quotients and subgroups of Baumslag-Solitar groups  
[1602.02445]: A Logspace Solution to the Word and Conjugacy problem of Generalized Baumslag-Solitar Groups  
[2210.14990]: On the space of subgroups of Baumslag-Solitar groups I: perfect kernel and phenotype  
[2402.16128]: Direct and Inverse Problems in Baumslag-Solitar Group $BS(1,3)$  
[2405.09736]: On the conjugacy separability of ordinary and generalized Baumslag-Solitar groups  
[2406.03960]: Cohomological Separability of Baumslag--Solitar groups and Their Generalisations  
[1003.0148]: Metabelian groups with quadratic Dehn function and Baumslag-Solitar groups

Source: https://www.emergentmind.com/topics/baumslag-solitar-group