New Trends in Geometric Group Theory and Geometric Topology

Dear Colleagues,

Geometric group theory has become a central and dynamic field at the intersection of geometry and low-dimensional topology, tracing its origins to John Stallings’ seminal work on the ends of groups. Since then, a sequence of foundational contributions—Gromov’s theory of hyperbolic groups, Bass–Serre theory of group actions on trees, the introduction of Outer space by Culler and Vogtmann, Bestvina–Feighn’s theory of limit groups, and the JSJ decomposition frameworks of Sela–Rips and Guirardel—has shaped the modern landscape of the field and fueled its rapid development over the past several decades.

These advances are deeply rooted in the rich and mature theory of geometric topology. Following the resolution of Thurston’s Geometrization Conjecture and the Virtual Fibering Conjecture, geometric topology has reached a high point, extending its influence to a broad range of mathematical disciplines. Recent developments have pushed into higher-dimensional topology and symplectic geometry, particularly through the successes of Heegaard Floer theory and its applications in symplectic settings.

A major focus of geometric group theory is the study of finitely generated groups through a geometric lens—a perspective grounded in the well-established theory of compact manifolds in geometric topology. Inspired by the Milnor–Schwarz Lemma, this approach introduces concepts such as quasi-isometry, which captures the coarse geometric structure of groups and reveals deep connections between their algebraic and geometric properties.

In parallel, the study of topologized groups—such as profinite groups, which arise naturally from topological covering theory—introduces a different layer of structure and rigidity. This notion of rigidity, echoing the local-to-global (Hasse) principles in number theory, has proven essential in understanding the finer structure of groups. Other key concepts, such as group growth (epitomized by Gromov’s polynomial growth theorem) and amenability (especially concerning invariant measures), further illustrate the intricate interplay among geometry, algebra, and analysis.

This Special Issue on “New Trends in Geometric Group Theory and Geometric Topology” is dedicated to showcasing innovative and foundational research at the forefront of these intertwined disciplines. We welcome high-quality submissions that present original results of broad interest. While the scope of the Special Issue is wide, priority will be given to contributions that align closely with the central themes and keywords listed.

I look forward to receiving your contributions.

Dr. Yong Hou
Prof. Dr. Peter B. Shalen
Guest Editors

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• JSJ decomposition
• profinite rigidity
• infinite packing
• embedding of groups
• outer space and outer limit
• group growth
• quasi-isometry
• hierarchical hyperbolic
• acylindrical hyperbolic
• amenability
• complex of groups
• cubulations
• higher rank groups and higher Teichmuller theory
• thin group
• Heegaard Floer theory