Growth of Approximate Groups in Hyperbolic Groups

Abstract: We prove a growth dichotomy for infinite approximate groups, and more generally approximate semigroups, in hyperbolic groups. If (G) is a finitely generated hyperbolic group and (A\subseteq G) is infinite with [ A^2\subseteq AX ] for some finite (X\subseteq G), then either (\langle A\rangle) is virtually cyclic, or (A) has positive exponential growth in the ambient word metric. We also introduce a product-growth criterion for the existence of growth rates of approximate semigroups. The criterion applies to hyperbolic groups: if (G) is hyperbolic with finite generating set (S), then there is a constant (c_{G,S}>0) such that [ |UV| \geq c_{G,S}\,\frac{|U||V|}{n+k+1}, \qquad U\subseteq B_n,\; V\subseteq B_k. ] The linear loss is optimal in order whenever (G) contains an element of infinite order. In the free group with its standard generating set one may take (c_{G,S}=1/4). We also prove that, in a free group, if (U\subseteq S_n) and (V\subseteq S_k), then [ |UV|\geq \left(\frac{2}{3}+\frac{1}{3\cdot 4^{\min{n,k}}}\right)|U||V|, ] and this constant is sharp for all (n,k).

• The paper demonstrates that any infinite approximate semigroup in a hyperbolic group is either virtually cyclic or exhibits positive exponential growth in the word metric.
• It employs a quantitative Schottky-type argument to derive sharp product-set lower bounds, with optimal estimates provided in free groups.
• The findings bridge additive combinatorics and geometric group theory, offering actionable insights for analyzing growth phenomena in nonabelian settings.

Growth Dichotomy and Product Set Expansion for Approximate Groups in Hyperbolic Groups

Overview

This paper establishes sharp dichotomies and product-set lower bounds for infinite approximate groups and semigroups in finitely generated hyperbolic groups. The principal result is a growth dichotomy: any infinite approximate semigroup $A \subseteq G$, with $G$ hyperbolic and $A^2 \subseteq AX$ for finite $X\subset G$, satisfies that either its ambient subgroup $\langle A \rangle$ is virtually cyclic or $A$ exhibits positive exponential growth in the word metric. The analysis also introduces and optimizes product-set estimates for finite subsets in balls of hyperbolic groups, with particular sharpness in the free group case, and connects these to general product-growth and Rapid Decay (RD) properties.

Main Theoretical Contributions

Hyperbolic Growth Dichotomy

Let $G$ be a finitely generated hyperbolic group and $A \subseteq G$ an infinite approximate semigroup. The main result asserts:

1. Dichotomy: If $A^2 \subseteq AX$ for finite $X$, then either $G$0 is virtually cyclic, or $G$1 has strictly positive exponential growth in the word metric.
2. Minimal Assumptions: No structural or geometric/lattice hypothesis on $G$2 is needed beyond one-sided small doubling; symmetry and even finite generation of $G$3 are not required.
3. Quantitative Schottky Argument: The proof leverages a quantitative separation technique akin to Schottky groups: constructing many long products of elements in $G$4 and showing, via hyperbolicity, that their images are sufficiently separated to withstand the accumulative error in the approximate group structure, thereby ensuring exponential growth.

Existence of Growth Rates via Product Expansion

The paper isolates a product-set lower bound property, referred to as large product growth:

• There exists a polynomial $G$5 such that for all finite $G$6:

$G$7

This quantitative lower bound allows the establishment of the well-defined exponential growth rate for every non-empty approximate semigroup $G$8:

$G$9

exists, with explicit summability requirements on the "shrinkage" function controlling product-set collapse.

Sharp Product-Set Lower Bounds in Hyperbolic and Free Groups

Two optimal estimates for ordinary products are obtained:

• In free groups: For finite sets $A^2 \subseteq AX$0,

$A^2 \subseteq AX$1

and for $A^2 \subseteq AX$2 (spheres), the exact bound is

$A^2 \subseteq AX$3

which is tight for every $A^2 \subseteq AX$4.

• In hyperbolic groups: For $A^2 \subseteq AX$5, there is a constant $A^2 \subseteq AX$6 depending on $A^2 \subseteq AX$7 and the generating set $A^2 \subseteq AX$8 so that

$A^2 \subseteq AX$9

These results are not only sharp in their order of dependence on radii but, in the free group, determine the optimal constants.

Consequences for Groups with Rapid Decay

A general result proved is that any finitely generated group with RD admits a polynomial product-set lower bound (via Sapir's "Rapid Expansion" property). Thus, all approximate groups in hyperbolic groups, groups of polynomial growth, and others with RD have well-defined exponential growth rates for their counting functions.

Technical Insights and Proof Structure

• Growth dichotomy: The dichotomy is obtained via a Schottky-type construction, exploiting the large translation length of elements within $X\subset G$0 ensured by non-elementarity. In the virtually cyclic (elementary) case, all elements essentially stabilize a boundary point.
• Product-set expansion: The core quantitative combinatorial innovation is a layer-peeling argument that reduces product-set estimates to optimizing overlap in "heavy" prefix or suffix classes. In the free group case, an explicit combinatorial optimization yields the tight bounds.
• Ambiguity and shrinkage: The ambient group's geometry controls potential collapse in the multiplication map; in hyperbolic groups, fibering is carefully bounded in Gromov product space.

Implications and Future Directions

• For additive combinatorics and geometric group theory, the paper bridges the local doubling structure to global asymptotic growth and provides explicit, calculable thresholds between structured and "chaotic" (expanding) behavior for approximate semigroups in negatively curved groups.
• The results underline that, in hyperbolic settings (and more generally for groups with large product growth), there are no "intermediate" behaviors: any infinite approximate group is either polynomially or exponentially large.
• The characterization of optimal product expansion rates opens up precise quantitative analysis of combinatorial growth phenomena in nonabelian groups.

Speculative Directions

• The approach could be extended to other classes of groups, e.g., acylindrically hyperbolic, mapping class groups, or linear groups, potentially with modifications to account for less rigid boundary behavior.
• Analogues in measured groupoids, approximate subgroups in Lie groups, or connections to expander graphs and spectral gaps in group actions could be explored using the established framework.
• For algorithmic group theory, the sharp product bounds may inform efficient algorithms for membership and growth rate estimation in non-commutative group settings.

Conclusion

This work rigorously delineates the structural and asymptotic growth properties of infinite approximate groups in hyperbolic groups, combining geometric, combinatorial, and analytic methods to derive sharp qualitative and quantitative results. The dichotomy between virtual cyclicity and exponential growth, together with optimal product-set estimates, significantly clarifies the boundary between algebraic structure and expansion in negatively curved groups. The marriage of additive combinatorics and geometric group theory realized here provides a foundation for further developments in the analysis of approximate symmetries in non-abelian settings.

Reference: "Growth of Approximate Groups in Hyperbolic Groups" (2606.13632)