On boundaries of bicombable spaces

TL;DR

The paper develops two parallel EZ-structure constructions for groups acting geometrically on spaces with a $\mathfrak{ccc}$ geodesic bicombing and proves their equivalence, significantly broadening the boundary toolkit beyond CAT(0) and hyperbolic settings. It analyzes and compares boundary notions via geodesic rays and Gelfand duals, establishes a quasisymmetric boundary structure, and studies how product and cube-complex constructions interact with boundary topology. A key result shows non-uniqueness of boundaries in the injective setting, contrasting with rigidity in CAT(0) spaces, while the almost geodesic completeness result links boundary topology to coarse cohomology. The work further explores boundary behavior under group decompositions, abelian subgroups, and provides a suite of open questions about boundary rigidity, Helly groups, and higher-dimensional CCC-bicombable spaces, with potential implications for the Novikov conjecture and coarse geometry of groups.

Abstract

We initiate systematic study of EZ-structures (and associated boundaries) of groups acting on spaces that admit consistent and conical (equivalently, consistent and convex) geodesic bicombings. Such spaces recently drew a lot of attention due to the fact that many classical groups act `nicely' on them. We rigorously construct EZ-structures, discuss their uniqueness (up to homeomorphism), provide examples, and prove some boundary-related features analogous to the ones exhibited by CAT(0) spaces and groups, which form a subclass of the discussed class of spaces and groups.

Theorems & Definitions (62)

• Theorem 1
• Corollary 2
• Remark 3
• Theorem 4: Theorem \ref{['t:nonunique']}
• Theorem 5: Corollary \ref{['c:ccc']}
• Proposition 6: see Proposition \ref{['p:quasisym']}\ref{['p:quasisym5']}
• Proposition 7: Proposition \ref{['p:amalgamated']}
• Proposition 8: Proposition \ref{['p:sphereinbdry']}
• Theorem 9: Theorem \ref{['t:agcmt']}
• Theorem 10: Theorem \ref{['t:agcc3']}