Baumslag–Solitar Group

• Baumslag–Solitar groups are two-generator one-relator HNN-extensions defined by t a^m t⁻¹ = a^n, showcasing key features like amenability and non-Hopfian behavior.
• They possess rich subgroup and quotient structures with embedding criteria and residual properties tightly linked to the integer parameters m and n.
• Their algorithmic complexity is reflected in LOGSPACE-complete word and conjugacy problems, providing insights into geometric and combinatorial group phenomena.

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 $a^n$ (Kida, 2011, Weiß, 2016). 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 $$\mathrm{BS}(1,-1)\cong\mathbb{Z}\ast(\mathbb{Z}/2\mathbb{Z})$$ (Sokolov, 2024).
• 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 $t a t^{-1}=a^n$ (Cornulier et al., 2010).
• 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 (Weiß, 2016, Levitt, 2013).
• 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 (Sokolov, 2024, Cohen et al., 2024, Weiß, 2016).

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 (Levitt, 2013, Carderi et al., 2022).

• 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 (Levitt, 2013).
• 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 $G\twoheadrightarrow BS(m,n)$ (Levitt, 2013).
• 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 (Carderi et al., 2022).

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" (Sokolov, 2024, Cohen et al., 2024).

• 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 (Cohen et al., 2024).
• 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 (Sokolov, 2024).
• Cohomological Separability ("goodness"): The trichotomy for $\mathrm{BS}(m,n)$ (Cohen et al., 2024):

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 (McLaury, 2011):

• 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 (Weiß, 2016):

• 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 (Cornulier et al., 2010, Singh et al., 2024).

• 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 $\Gamma_n$ (Cornulier et al., 2010).
• 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 (Singh et al., 2024). 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 (Kida, 2011):

• 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.

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References:

(Kida, 2011): Invariants of orbit equivalence relations and Baumslag-Solitar groups (McLaury, 2011): Irreducible Representations of Baumslag-Solitar Groups (Levitt, 2013): Quotients and subgroups of Baumslag-Solitar groups (Weiß, 2016): A Logspace Solution to the Word and Conjugacy problem of Generalized Baumslag-Solitar Groups (Carderi et al., 2022): On the space of subgroups of Baumslag-Solitar groups I: perfect kernel and phenotype (Singh et al., 2024): Direct and Inverse Problems in Baumslag-Solitar Group $BS(1,3)$ (Sokolov, 2024): On the conjugacy separability of ordinary and generalized Baumslag-Solitar groups (Cohen et al., 2024): Cohomological Separability of Baumslag--Solitar groups and Their Generalisations (Cornulier et al., 2010): Metabelian groups with quadratic Dehn function and Baumslag-Solitar groups