Conjugacy in Discrete Subgroups

• Conjugacy in discrete subgroups is defined by the existence of an inner automorphism linking two elements or subgroups, partitioning them into equivalence classes.
• The discrete Heisenberg group illustrates how invariants, such as the unchanged horizontal pair and the modular central parameter, classify conjugacy classes.
• Algorithmic methods, including Stallings folding and JSJ decomposition, are used to decide conjugacy and assess stability in free, hyperbolic, and affine groups.

A conjugacy relation on discrete subgroups, broadly speaking, organizes elements or subgroups of a discrete group according to conjugation equivalence: two elements or subgroups are conjugate if they are related through an inner automorphism. This relation underpins the structure of many areas in algebraic and geometric group theory, especially in the context of discrete subgroups of classical and non-classical groups, and impacts problems such as classification, geometric representation, and algorithmic analysis.

1. Conjugacy in Discrete Groups: General Framework

In any group $G$, elements $x, y \in G$ are conjugate if there exists $g \in G$ such that $x = g y g^{-1}$. This yields the partition of $G$ into conjugacy classes. For discrete subgroups (i.e., groups with the discrete topology, such as lattices in Lie groups or finitely generated linear groups over discrete rings), understanding conjugacy classes is essential for representation theory, geometric topology, and the analysis of group actions.

The conjugacy relation is equally pivotal for subgroups: two subgroups $H, K \leq G$ are conjugate in $G$ if $\exists g \in G$ such that $K = g H g^{-1}$.

The study of conjugacy relations involves both computation (providing complete invariants, classifying conjugacy classes, or algorithms to decide conjugacy) and geometric or homotopical invariants (in relation to the ambient space on which the group acts) (Martínez et al., 2020, Budylin, 2014).

2. Explicit Conjugacy Relations: The Discrete Heisenberg Group

A canonical example is the discrete Heisenberg group $H_\mathbb{Z}$: $H_\mathbb{Z} = \{ (x, y, z) \in \mathbb{Z}^3 \}$ with the group law

$$(x, y, z) \cdot (x', y', z') = (x + x',\; y + y',\; z + z' + x y').$$

The conjugation formula, for $u = (x, y, z)$, $v = (x', y', z')$, is

$u^{-1} v u = (x', y', z' + x y' - x' y).$

This explicitly exhibits the conjugacy invariants: the "horizontal" pair $(x', y')$ is unchanged under conjugation, and the central parameter $z'$ shifts by the symplectic pairing $\omega((x, y), (x', y')) = x y' - x' y$. As $(x, y)$ varies over $\mathbb{Z}^2$, the possible shifts form the subgroup $d \mathbb{Z}$, where $d = \gcd(x', y')$. The invariants are thus $(x', y')$ and $z' \bmod d$ (Budylin, 2014).

The classification theorem for conjugacy in $H_\mathbb{Z}$ states: $(x, y, z)$ and $(x', y', z')$ are conjugate if and only if $(x, y) = (x', y')$ and $z \equiv z' \mod d$, with $d = \gcd(x, y)$. This yields a bijection between conjugacy classes and

$\mathbb{Z}^2 \times \mathbb{Z}/d\mathbb{Z}.$

3. Algorithmic Properties and Conjugacy Stability in Discrete Subgroups

Algorithmic questions about conjugacy—deciding if two elements (or subgroups) are conjugate—are central in combinatorial and geometric group theory. In the context of discrete subgroups, such as free groups, hyperbolic groups, and their subgroups, the property of conjugacy stability is crucial.

A subgroup $H < G$ is conjugacy stable in $G$ if $x, y \in H$ and $x$ is conjugate to $y$ in $G$ implies they are already conjugate in $H$.

Summary of key techniques:

• Free groups: Stallings foldings and graph-theoretic intersections.
• Torsion-free hyperbolic groups: Fellow-traveling quasi-geodesics with bounded search in the Cayley graph; intersections are typically infinite cyclic.
• Limit groups: JSJ decompositions and centralizer/coset-graph methods.

In each setting, the problem of conjugacy stability reduces to checking the structure of a finite collection of double coset representatives and testing membership in their intersection subgroups, exploiting their cyclic or abelian nature (Martínez et al., 2020).

Group Type	Conj. Stability Criterion	Algorithmic Ingredient
Free	Cyclic intersection in double cosets	Stallings graph folding
Torsion-free hyperbolic	Cyclic intersection (bounded by quasi-convexity constants)	Dehn algorithm, quasi-geodesics
Limit groups	Abelian intersection in finite families	JSJ decomposition, coset graphs

4. Conjugacy in Lattices and Euclidean Isometry Groups

For split discrete subgroups of the isometry group of Euclidean space (notably affine Coxeter and crystallographic groups), the conjugacy relation is controlled by the linear (spherical) part and the placement of the translation part within the underlying lattice.

Any element $h = t^\lambda h_0$ (translation by $\lambda$ and linear part $h_0$) has conjugacy class

$[h]_H = \bigcup_{u \in H_0} t^{u(\lambda + \Mod(h_0))}\, (u h_0 u^{-1}),$

where $\Mod(h) = (h - \Id)L_H$ captures how the translation part sits relative to the action of $h_0$ on the lattice.

The geometry of conjugacy classes is thus governed by the "move-set" $\Mov(h)$ and "fix-set" $\Fix(h)$, and their intersection with the lattice $L_H$ (Milićević et al., 2024). Affine Coxeter groups provide examples where these sets correspond to the affine structure of root hypertori and associated reflections.

5. Local Versus Global Conjugacy: Cohomological and Counting Criteria

A further refinement distinguishes between global conjugacy (equality up to a single conjugator) and local conjugacy (bijection of subgroups so each individual element is conjugate, but not necessarily all through a single conjugator). In finite and arithmetic group settings, local conjugacy can be detected via class functions:

For subgroups $H_1, H_2 \leq G$, they are locally conjugate if and only if for all conjugacy classes $C$ in $G$, $|H_1 \cap C| = |H_2 \cap C|$. This is a subtle equivalence, particularly relevant in $\mathrm{GL}_2(\mathbb{Z}/p^2 \mathbb{Z})$, where non-conjugate, locally conjugate subgroups arise in split-Cartan and certain Borel cases (Kim, 2017).

6. Chains of Discrete Subgroups and Homotopical Invariants

The classification of conjugacy classes of chains of discrete subgroups, particularly $p$-toral subgroups in compact Lie groups, is central in the study of fusion systems and classifying spaces. Given a maximal discrete $p$-toral subgroup $S$ in a compact Lie group $G$, and the associated fusion system $\mathcal{F}_S(G)$, two chains are conjugate if they are related by an inner automorphism. Taking closures maps discrete chains to continuous $p$-toral chains, and this process induces an injective, and sometimes bijective, map on conjugacy classes (Belmont et al., 2021).

A fundamental result is that the inclusion of normalizers along such chains induces a mod $p$ equivalence on classifying spaces, ensuring that discrete conjugacy data suffices to recover the full normalizer decomposition in the homotopy theory of $G$.

7. Conjugacy Problem for Subdirect Products in Hyperbolic Settings

For subdirect products of torsion-free hyperbolic groups, the decision problem for conjugacy is reducible to a uniform algorithm for membership in cyclic subgroups of a finitely presented quotient group. The solvability of the conjugacy problem in the discrete subdirect product $P < G_1 \times G_2$ is equivalent to the recursive computability of the so-called "rel-cyclics" Dehn function in the quotient $Q = G_1/(P \cap G_1)$.

A pivotal technique is perturbing elements to avoid proper powers, which is possible due to the rigidity of centralizers in hyperbolic groups (they are infinite cyclic in the torsion-free context). This alignment precisely characterizes when conjugacy can be algorithmically decided in such complex discrete settings (Bridson, 7 Jul 2025).

---

The conjugacy relation on discrete subgroups, across diverse group-theoretic and geometric settings, thus manifests as a central organizing principle: its explicit algebraic invariants, algorithmic decidability, geometric interpretation, and cohomological implications govern classification, computation, and the architecture of higher invariants in group theory.