Core Concepts

Non-elementary hyperbolic groups exhibit finite index rigidity, meaning isomorphic finite-index subgroups always share the same index. This property stems from a connection between a hyperbolic group's topological complexity and the covolume of its actions on hyperbolic spaces.

Abstract

**Bibliographic Information:**Lazarovich, N. (2024). Finite index rigidity of hyperbolic groups. arXiv preprint arXiv:2302.04484v3.**Research Objective:**This paper investigates the relationship between the topological complexity of a hyperbolic group and the covolume of its actions on hyperbolic spaces, ultimately aiming to prove the finite index rigidity of non-elementary hyperbolic groups.**Methodology:**The author utilizes Mineyev's rational bicombing on hyperbolic graphs, drawing inspiration from Delzant's foliation techniques and Rips-Sela stable cylinders. By constructing weighted singular patterns on simplicial complexes associated with hyperbolic groups, the study establishes inequalities relating covolume, complexity, and index.**Key Findings:**The paper demonstrates that the topological complexity of a finite index subgroup within a hyperbolic group is linearly proportional to its index. This finding is a consequence of a broader result linking the size of the quotient of a free cocompact action of a hyperbolic group on a graph to the minimal cell count in a simplicial classifying space for the group.**Main Conclusions:**The central conclusion is that non-elementary hyperbolic groups are inherently finite index rigid. This implies that any two isomorphic finite-index subgroups within a non-elementary hyperbolic group must possess the same index.**Significance:**This research significantly contributes to geometric group theory by providing a novel approach to understanding the structure and properties of hyperbolic groups. The established connection between topological complexity and index has profound implications for studying group actions and their associated geometric invariants.**Limitations and Future Research:**The paper primarily focuses on one-ended hyperbolic groups. Exploring similar relationships in the context of groups with multiple ends could be a potential avenue for future research. Additionally, investigating the implications of finite index rigidity in broader geometric and topological settings would be of interest.

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by Nir Lazarovi... at **arxiv.org** 10-15-2024

Deeper Inquiries

The concept of finite index rigidity can be explored in the broader context of groups acting on various spaces, not just hyperbolic ones. Here's how:
General Idea: The core idea remains similar: we aim to connect the "complexity" of a group to the "size" of its action on a space. If a group has isomorphic finite-index subgroups of different indices, it suggests a certain "flexibility" in its action, hinting that the space might not be "rigid" enough to distinguish these subgroups.
Beyond Hyperbolic Spaces:
CAT(0) spaces: Many results from the hyperbolic setting have analogues in CAT(0) spaces. For instance, one could investigate if the covolume of a group acting properly and cocompactly on a CAT(0) space can be bounded in terms of some group-theoretic complexity invariant.
Symmetric spaces and lattices: The study of lattices in Lie groups is intimately connected to finite index rigidity. Lattices in higher-rank Lie groups exhibit strong rigidity properties, including finite index rigidity. This rigidity stems from the rich geometric and algebraic structure of these spaces.
Coarse geometry: Finite index rigidity can be viewed through the lens of coarse geometry. If two groups are quasi-isometric and one is finite index rigid, the other often inherits some form of rigidity, though not necessarily finite index rigidity in its full strength.
Key Challenges:
Finding suitable complexity invariants: The choice of a complexity invariant that captures the relevant information about the group's action is crucial. This choice depends heavily on the specific type of space being considered.
Relating complexity to the action: Establishing a precise relationship between the chosen complexity invariant and the "size" of the action (e.g., covolume, growth rate) is essential. This often requires developing new geometric and combinatorial techniques tailored to the specific space.

Yes, counterexamples to finite index rigidity definitely exist among non-hyperbolic groups. Here are some characteristics and examples:
Characteristics Favoring Non-Rigidity:
High flexibility: Groups with a high degree of "flexibility" in their subgroups are more likely to violate finite index rigidity. This flexibility might manifest as:
Many subgroups of finite index: Groups with a rich lattice of finite index subgroups have more "room" for isomorphic subgroups of different indices.
Amenability: Amenable groups, which exhibit a certain "averaging" property, often have less rigid subgroup structure.
Lack of geometric invariants: Groups lacking strong geometric invariants that control their actions are more prone to non-rigidity.
Examples:
Infinite abelian groups: As mentioned in the context, the infinite cyclic group Z is not finite index rigid. More generally, any infinite abelian group fails to be finite index rigid.
Groups with infinitely many ends: While groups with infinitely many ends are finite index rigid, they can have isomorphic subgroups of different infinite indices.
Baumslag-Solitar groups: Certain Baumslag-Solitar groups, defined by presentations like BS(m,n) = ⟨a, b | bamb-1 = an⟩, provide examples of non-hyperbolic groups that are not finite index rigid.

This research on finite index rigidity for hyperbolic groups has several implications for the broader study of finitely generated groups:
Deeper Understanding of Rigidity:
Geometric and algebraic interplay: It highlights the intricate interplay between geometric properties of groups (hyperbolicity, actions on spaces) and their algebraic structure (subgroup structure, isomorphism classes).
New avenues for investigating rigidity: It motivates exploring finite index rigidity and related properties in other classes of groups, such as CAT(0) groups, relatively hyperbolic groups, and mapping class groups.
Connections to Growth:
Growth sensitivity to index: Finite index rigidity implies that the "complexity" of a group, as measured by invariants like the ones discussed in the context, is sensitive to the index of finite index subgroups. This sensitivity suggests a connection between the growth rate of a group and its finite index subgroups.
Exploring growth variations: It encourages investigating how growth rates of groups within a particular class can vary and whether this variation can be linked to the presence or absence of finite index rigidity.
Broader Impact:
New tools and techniques: The methods developed in this research, such as using globally stable bicombings and analyzing weighted singular patterns, could potentially be adapted to study other aspects of group structure and growth.
Classifying groups: Understanding rigidity properties like finite index rigidity contributes to the broader program of classifying finitely generated groups by understanding their algebraic and geometric invariants.

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