Acylindrically Hyperbolic Groups

• Acylindrically hyperbolic groups are defined as groups that admit non-elementary acylindrical actions on hyperbolic metric spaces, ensuring strong properness conditions.
• Recent studies extend model-theoretic results to these groups using advanced techniques like test sequences, small-cancellation arguments, and limit group analysis.
• Verbal subgroups in acylindrically hyperbolic groups exhibit infinite width, reflecting the groups' complex and rich algebraic structures.

Acylindrically hyperbolic groups constitute a major unifying class within geometric group theory, lying strictly between the class of non-elementary Gromov hyperbolic groups and much wider classes like mapping class groups and $\mathrm{Out}(F_n)$. A group $G$ is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a (Gromov) hyperbolic metric space—that is, a group action with strong properness conditions off the diagonal, ensuring that only finitely many group elements can move any pair of far-apart points within a uniform distance. This class, introduced and systematically developed by Osin and collaborators, is now central to structural and model-theoretic advances in infinite group theory.

1. Definitions and Core Characterizations

Let $(X,d)$ be a Gromov–hyperbolic metric space. An isometric action $G \curvearrowright X$ is acylindrical if for every $\varepsilon > 0$ there exist $R,N<\infty$ such that for any $x,y\in X$ with $d(x,y)\ge R$,

$$\bigl|\{\,g\in G : d(x,gx)\le \varepsilon,\ d(y,gy)\le \varepsilon\}\bigr| \le N.$$

The action is non-elementary if $G$ contains two independent loxodromic elements (with disjoint fixed-points on the boundary).

A group $G$ is acylindrically hyperbolic if it admits a non-elementary acylindrical action on some hyperbolic space $X$. Several equivalent characterizations are available:

• $G$ is not virtually cyclic and has such an action.
• $G$ admits a loxodromic element satisfying the WPD (“weak proper discontinuity”) property.
• $G$ contains a proper infinite hyperbolically embedded subgroup.

Standard examples include non-elementary Gromov hyperbolic groups, non-elementary relatively hyperbolic groups, mapping class groups, most $3$-manifold groups, many right-angled Artin groups, certain Artin-Tits groups, and any nontrivial free product except $\mathbb{Z}/2*\mathbb{Z}/2$ (Osin, 2013, Calvez, 2020, Przytycki et al., 2015).

2. Model-Theoretic Properties and Elementary Embeddings

Classical results of Merzlyakov on implicit function theorems for free groups have recently been extended to the full class of acylindrically hyperbolic groups. If $G$ is acylindrically hyperbolic and $E(G)$ its unique maximal finite normal subgroup, André and Fruchter show that the natural embedding of $G$ into the HNN extension

$$\Gamma = G \dot{*}_{E(G)} = \langle G, t \mid [t, g]=1 \ \forall g \in E(G) \rangle$$

is an $\exists\forall\exists$-elementary embedding. Consequently, $G$ and $\Gamma$ share the same $\forall\exists$-theory: every $\forall\exists$-sentence true in $G$ is true in $\Gamma$ (André et al., 2020). The significance of this result is the transfer of logical structure between $G$ and a substantially larger group into which $G$ embeds.

The proof requires advanced machinery: the construction of test sequences of homomorphisms for equations, small-cancellation arguments to prevent collapse to solvable or virtually abelian cases, and limit group techniques (actions on $\mathbb{R}$-trees and the Rips machine) to analyze the structure of solutions. Notably, equational Noetherianity is not required: relative approximations and a generalized shortening argument allow retractions to be found without this hypothesis.

3. Triviality of Positive Theory

One key corollary is that all acylindrically hyperbolic groups have trivial positive theory in the sense of first-order logic: every positive sentence (in the group language, built using only conjunction, disjunction, and the quantifiers $\forall$ and $\exists$—no negations or inequalities) holding in $G$ holds in all groups. This result settles a conjecture of Casals-Ruiz, Garreta, and de la Nuez, and extends the classical situation for free groups (André et al., 2020).

This is seen via two routes:

1. Through the elementary equivalence between $G$ and $\Gamma$, which surjects onto a free group; since every positive sentence true in a non-abelian free group holds in all groups, the theory collapses.
2. Directly via the generalized Merzlyakov theorem: solutions to positive $\forall\exists$-sentences in $G$ can always be encoded by retractions to free factors, ensuring absolute generality.

4. Width of Verbal Subgroups

A further group-theoretic outcome is that verbal subgroups of acylindrically hyperbolic groups have infinite width except for the obvious cases. For a word $w(x_1,...,x_k)\in F_k$, if the greatest common divisor of exponent sums $d(w)\neq 1$, then the verbal subgroup $$w(G)=\langle w(\mathbf{g})\mid \mathbf{g}\in G^k\rangle$$ has infinite width: no uniform bound exists on the number of $w$-values required to express every element of $w(G)$. This extends prior results for non-abelian free groups to the full acylindrically hyperbolic class, confirming and generalizing theorems of Bestvina–Bromberg–Fujiwara (André et al., 2020).

The logical mechanism is that, for $w$ as above, the absence of a positive sentence bounding the product length for $w(G)$ implies infinite width by a dichotomy from trivial positive theory. This is reinforced by examples of Segal and Rhemtulla for free groups.

5. Technical Scheme: Test Sequences and Limit Group Analysis

The proof approach synthesizes several advanced techniques:

• Test sequences: Constructed from small-cancellation arguments in quasi-convex free subgroups, ensuring non-trivial solution growth and resistance to collapse under equivalence.
• Limit groups: By taking ultralimits of Cayley graphs rescaled by displacement of test sequences, one obtains actions on real trees. The Rips machine and JSJ decompositions are used to analyze the structure and extract retractions.
• Shortening argument: In absence of equational Noetherianity, a relative approximation is constructed that captures the limiting behavior of test sequences, with modular automorphisms used to enforce minimality and prevent unwanted foldings.

These methods adapt and generalize prior work on free and hyperbolic groups to the broader class of acylindrically hyperbolic groups.

6. Context, Impact, and Broader Significance

The categorical triviality of positive theory for acylindrically hyperbolic groups is highly nontrivial, emphasizing that such groups are, from the standpoint of positive first-order logic, indistinguishable from non-abelian free groups or even their free products with $\mathbb{Z}$. This result has deep implications for the algebraic geometry of equations over groups, the structure of limit groups, and the model theory of infinite groups (André et al., 2020). The infinite width of verbal subgroups further signals the complexity and abundance of word values in these groups, precluding the kind of algebraic rigidity seen in settings like finite or linear groups.

The techniques established in this area are broadly influential in extending the reach of small cancellation, ultralimit analysis, and geometric group-theoretic model theory. They combine logical, combinatorial, and geometric insights, reflecting the methodological heterogeneity characteristic of contemporary geometric group theory.

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References

• Formal solutions and the first-order theory of acylindrically hyperbolic groups (André et al., 2020)
• Acylindrically hyperbolic groups (Osin, 2013)
• A note on acylindrical hyperbolicity of Mapping Class Groups (Przytycki et al., 2015)
• Euclidean Artin-Tits groups are acylindrically hyperbolic (Calvez, 2020)