Core Concepts

The mapping class group of a locally finite graph Γ, denoted Map(Γ), acts faithfully on the sphere complex S(MΓ) of the associated doubled handlebody MΓ. This action is induced by a short exact sequence that splits topologically, giving an isomorphism Map(MΓ) ∼= Twists(MΓ) ⋊ Map(Γ), where Twists(MΓ) is a compact abelian group of sphere twists.

Abstract

The paper studies the mapping class group Map(Γ) of a locally finite graph Γ, as defined by Algom-Kfir and Bestvina. The main result is that Map(Γ) acts faithfully on the sphere complex S(MΓ) of the associated doubled handlebody MΓ. This is achieved by proving a short exact sequence 1 → Twists(MΓ) → Map(MΓ) → Map(Γ) → 1, where Twists(MΓ) is a compact abelian group of sphere twists. Furthermore, this short exact sequence splits topologically, giving an isomorphism Map(MΓ) ∼= Twists(MΓ) ⋊ Map(Γ).

Along the way, the paper obtains several other results:

- It precisely determines the group Twists(MΓ), showing it is topologically isomorphic to Πrk(Γ)

i=1 Z/2. - It studies the connectivity properties of S(MΓ) and its subcomplexes, extending previous results.
- It investigates the action of Map(Γ) on the nonseparating sphere complex Sns(MΓ) for finite rank graphs, showing the action has elements with positive translation length.
- It defines an "Outer space" O(Γ) for finite rank graphs and proves it is contractible.
- It translates results about pure mapping class groups of graphs to pure mapping class groups of doubled handlebodies.
- It studies a class of "translatable graphs" and shows their mapping class groups are coarsely bounded generated and quasi-isometric to the sphere complex.

To Another Language

from source content

arxiv.org

Stats

The rank of the fundamental group of Γ is denoted rk(Γ).
The group Twists(MΓ) is topologically isomorphic to Πrk(Γ)
i=1 Z/2.

Quotes

"The mapping class group Map(Γ) of a locally finite graph Γ, denoted Map(Γ), acts faithfully on the sphere complex S(MΓ) of the associated doubled handlebody MΓ."
"The short exact sequence 1 → Twists(MΓ) → Map(MΓ) → Map(Γ) → 1 splits topologically, giving an isomorphism Map(MΓ) ∼= Twists(MΓ) ⋊ Map(Γ) as topological groups."

Key Insights Distilled From

by Brian Udall at **arxiv.org** 10-03-2024

Deeper Inquiries

The faithful action of the mapping class group Map(Γ) on the sphere complex S(MΓ) has significant implications for both the algebraic and geometric properties of Map(Γ).
From an algebraic perspective, the existence of a faithful action indicates that Map(Γ) can be effectively studied through its action on S(MΓ). This means that the elements of Map(Γ) can be represented as transformations of the sphere complex, allowing for the application of combinatorial and geometric techniques to analyze the structure of the group. The short exact sequence established in Theorem 1.1, which relates Map(Γ) to the mapping class group of the doubled handlebody Map(MΓ), suggests that understanding the kernel, Twists(MΓ), and its action on S(MΓ) can yield insights into the structure of Map(Γ). Specifically, the compact abelian nature of Twists(MΓ) implies that Map(Γ) retains a level of complexity that can be explored through its interactions with the sphere complex.
Geometrically, the action of Map(Γ) on S(MΓ) provides a framework for understanding the coarse geometry of the mapping class group. The results indicating that elements of Map(Γ) can have positive translation lengths in certain subcomplexes of S(MΓ) suggest that the group exhibits nontrivial geometric behavior, akin to the behavior observed in the mapping class groups of finite-type surfaces. This can lead to the conclusion that Map(Γ) possesses properties such as infinite diameter and quasi-isometric embeddings, which are crucial for understanding the geometric structure of the group. Furthermore, the connection to the sphere complex allows for the exploration of the topological features of the underlying graph Γ, enhancing our understanding of how the graph's structure influences the mapping class group's properties.

Yes, the techniques developed in this paper can be extended to study the mapping class groups of other infinite-type spaces, including non-locally finite graphs and infinite-type surfaces. The foundational concepts introduced, such as the sphere complex S(MΓ) and the associated mapping class groups, provide a robust framework that can be adapted to various infinite-type settings.
For non-locally finite graphs, the key challenge lies in the lack of compactness and the potential for more complex end structures. However, the methods of analyzing the action of mapping class groups on associated complexes can still be applied. By constructing appropriate analogs of the sphere complex for these graphs, one can investigate the algebraic and geometric properties of their mapping class groups. The results regarding the faithful action of Map(Γ) on S(MΓ) can serve as a model for understanding similar actions in non-locally finite contexts, potentially leading to new insights into their structure and behavior.
In the case of infinite-type surfaces, the existing literature on big mapping class groups provides a rich backdrop for applying the techniques from this paper. The parallels between the sphere complex of a doubled handlebody and the curve complex of an infinite-type surface suggest that similar results regarding the action of mapping class groups can be obtained. The exploration of translation lengths, quasi-isometry, and the structure of the kernel in the context of infinite-type surfaces can yield valuable information about their mapping class groups, analogous to the findings for locally finite graphs.

The coarse geometric properties of Map(Γ) and Map(MΓ) are intricately linked to the structure of the underlying graph Γ, extending beyond the specific case of "translatable graphs." The rank of the graph, the number of ends, and the connectivity properties all play crucial roles in determining the geometric behavior of the mapping class groups.
For instance, the rank of the graph Γ directly influences the complexity of the mapping class group. Higher ranks typically correspond to more intricate interactions within the graph, leading to a richer structure in the mapping class group. The presence of multiple ends can also affect the coarse geometry, as it introduces additional pathways and potential for nontrivial actions on the sphere complex. The results indicating that the action of Map(Γ) on certain subcomplexes has elements with positive translation length suggest that the underlying graph's topology significantly impacts the geometric properties of the mapping class group.
Moreover, the connectivity of the graph and its end structure can lead to variations in the asymptotic behavior of the mapping class groups. For example, graphs with infinite type and complex end structures may exhibit properties such as infinite diameter and quasi-isometric embeddings, similar to those observed in the mapping class groups of infinite-type surfaces. This relationship highlights the importance of understanding the graph's topology to draw conclusions about the geometric properties of its associated mapping class group.
In summary, the coarse geometric properties of Map(Γ) and Map(MΓ) are deeply influenced by the structural characteristics of the underlying graph Γ, and exploring these connections can yield a more comprehensive understanding of the mapping class groups in various contexts.

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