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This mathematics research paper presents a method for efficiently generating generalized loxodromic elements within subgroups of graph products using positive words, and explores its implications for the growth of subsets within these groups.

Abstract

**Bibliographic Information:**Fioravanti, E., & Kerr, A. (2024). Short positive loxodromics in graph products.*arXiv preprint arXiv:2410.06751v1*.**Research Objective:**The paper investigates the generation of generalized loxodromic elements in subgroups of graph products using positive words, aiming to establish a relationship between word length and the existence of such elements.**Methodology:**The authors utilize the properties of graph products, particularly their actions on Bass-Serre trees and contact graphs, to analyze the structure of subgroups and the behavior of elements within these groups. They employ concepts like essential support, stable support, and the bounded creasing property to construct elements with desired properties.**Key Findings:**The paper's main result (Theorem A) demonstrates the existence of a bound, dependent on the graph structure, on the length of positive words required to generate regular or strongly irreducible elements within subgroups of graph products. This finding has significant implications for understanding the growth of subsets in these groups.**Main Conclusions:**The authors conclude that graph products of groups with strong product set growth properties inherit these properties. Additionally, they establish that the set of growth rates for a specific class of subgroups within any graph product of equationally Noetherian groups is well-ordered.**Significance:**This research contributes significantly to the field of geometric group theory, particularly in the study of acylindrically hyperbolic groups and their growth properties. The findings have implications for understanding the structure and dynamics of these groups, with potential applications in areas like geometric topology and the study of algorithms on groups.**Limitations and Future Research:**The paper primarily focuses on finite graphs and specific types of elements within graph products. Future research could explore extending these results to infinite graphs or investigating the generation of other types of elements. Additionally, exploring the applications of these findings in related areas like computational group theory and the study of group actions on other geometric structures could be promising avenues for further investigation.

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by Elia Fiorava... at **arxiv.org** 10-10-2024

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This paper focuses on generating short loxodromic elements in graph products and applies this to understand the growth of their subgroups. Extending these findings to more general groups requires addressing several challenges:
1. Identifying Suitable Decompositions and Actions:
Beyond Graph Products: Graph products have a natural structure that allows for decomposition into vertex groups and actions on associated Bass-Serre trees. Generalizing to other groups requires finding analogous decompositions that capture essential structural information. Potential candidates include:
Relative Hyperbolic Groups: These groups decompose into parabolic subgroups and have actions on relative hyperbolic spaces.
HNN extensions and Amalgamated Free Products: These fundamental constructions have associated Bass-Serre trees that could be analyzed.
Groups acting on CAT(0) cube complexes: These groups often admit useful decompositions and actions that might be leveraged.
Finding Acylindrical Actions: The paper heavily relies on the existence of acylindrical actions on hyperbolic spaces or quasi-trees. For more general groups, the existence and properties of such actions need to be investigated.
2. Generalizing the Notion of Support:
Abstracting Support: The concept of support in graph products is closely tied to the vertex groups. A more abstract notion of support, perhaps based on the geometry of an action or a suitable decomposition, would be needed for broader applicability.
3. Adapting Techniques for Combining Elements:
New Combination Methods: The paper develops techniques for combining elements to produce loxodromics with larger support. These techniques rely on the specific structure of graph products and their actions on trees. New methods for combining elements, potentially using different word properties or group actions, would need to be explored.
4. Analyzing Growth in Different Settings:
Beyond Product Set Growth: While the paper focuses on product set growth, other growth functions, such as word growth or growth of conjugacy classes, could be studied using similar approaches.
Potential Research Directions:
Investigate the growth of subgroups in relatively hyperbolic groups, focusing on the interplay between parabolic subgroups and the geometry of the relative Cayley graph.
Explore the use of CAT(0) cube complexes and their associated structures to understand the growth of subgroups in groups acting on them.
Develop a more general framework for combining elements in groups with suitable decompositions, potentially drawing inspiration from the techniques used in this paper for graph products.

Yes, there are potentially alternative methods for generating loxodromic elements in graph products beyond those explored in the paper. Here are some possibilities:
1. Exploiting Geometric Properties of Words:
Geodesic Words or Quasi-Geodesic Words: Instead of focusing on positive words, one could investigate whether geodesic words (or words with bounded distance from geodesics) in the Cayley graph of the graph product are more likely to yield loxodromic elements. The geometry of the Cayley graph might provide insights into how to construct such words.
Words with Specific Subword Properties: Analyzing words with particular subword patterns or avoiding certain subwords might lead to criteria for identifying loxodromic elements. This could involve studying normal forms in graph products and how they relate to the geometry of the associated actions.
2. Leveraging Different Group Actions:
Actions on CAT(0) Cube Complexes: Graph products act naturally on CAT(0) cube complexes. Analyzing the interplay between the combinatorics of the cube complex and the structure of the graph product could lead to new methods for finding loxodromic elements.
Actions on Buildings: Right-angled Artin groups, a special case of graph products, act on their associated right-angled buildings. The rich geometric structure of buildings might provide tools for constructing loxodromic elements.
3. Using Random Walks and Probabilistic Methods:
Random Walks on Graph Products: Analyzing random walks on graph products and the properties of words generated by these walks could provide insights into the prevalence of loxodromic elements. Probabilistic methods might be useful for proving the existence of short loxodromic elements with certain properties.
4. Exploring Connections with Formal Language Theory:
Word Problems and Automata: The word problem in graph products is solvable, and there are connections between formal language theory and the structure of groups. Investigating these connections might lead to alternative characterizations of loxodromic elements in terms of formal languages or automata.

The findings of this paper have interesting implications for the computational complexity of problems related to loxodromic elements and subgroup growth in graph products:
1. Efficiently Identifying Loxodromic Elements:
Positive Witness for Loxodromics: Theorem A provides a bound on the length of positive words that need to be checked to find a loxodromic element (if one exists). This could lead to more efficient algorithms for identifying loxodromic elements, as it restricts the search space of potential candidates.
2. Estimating Growth Rates:
Approximating Growth: The connection between short loxodromic elements and product set growth (Corollary D) suggests that it might be possible to approximate the growth rate of a subgroup by finding short loxodromic elements in it. This could be computationally advantageous compared to directly computing the growth function.
3. Deciding Algorithmic Properties of Subgroups:
Finiteness and Virtual Properties: The existence of short loxodromic elements can be used to decide certain properties of subgroups. For example, Corollary B implies that one can decide whether a finitely generated subgroup of a virtually special group is finite or virtually abelian by checking for short loxodromic elements.
Challenges and Open Questions:
Complexity of Finding Shortest Loxodromics: While the paper provides bounds on the length of positive words representing loxodromic elements, it's unclear how hard it is to find the shortest such word. This could be an interesting question for further investigation.
Practical Algorithms: Translating the theoretical results into practical algorithms for identifying loxodromic elements and analyzing growth requires careful consideration of data structures and implementation details.
Dependence on Parameters: The bounds in the paper depend on the graph Γ or the dimension of the graph product. Understanding the precise dependence on these parameters and whether it can be improved is crucial for practical applications.
Potential Applications:
Geometric Group Theory Software: These findings could be incorporated into software packages for computational geometric group theory, leading to more efficient algorithms for analyzing graph products.
Cryptanalysis: Some cryptographic protocols rely on the difficulty of certain group-theoretic problems. Understanding the complexity of finding loxodromic elements in graph products could have implications for the security of such protocols.

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