Posed in 1900, Hilbert’s fifth problem was solved in its classical form in the 1950s. Its solution was a landmark of twentieth-century mathematics. Beyond local compactness, the search for structure turns to the geometry of infinite-dimensional spaces.

Building on the programme initiated by Per Enflo in the late 1960s, this monograph combines the classical background with modern geometric methods and original results by the authors, published here for the first time.

Two problems guide the investigation: determining when the closed additive subgroup generated by a continuous path is a real linear subspace and finding geometric conditions that guarantee square roots in a group. A central theorem identifies the critical path exponent of every infinite-dimensional real Banach space as the reciprocal of its supremal Rademacher type. The book also establishes quantitative square-root theorems for commutative group laws on Banach spaces and extends the path theory to metric groups. Explicit constructions and counterexamples reveal why the critical exponent alone does not determine the behaviour at the endpoint.

Substantial preliminaries, detailed proofs of the principal results, and a dedicated chapter of open problems provide a route from foundational ideas to current research. The book is intended for researchers and advanced graduate students in functional analysis, Banach-space geometry, nonlinear geometry, and topological group theory.

Features

Connects the classical theory of Hilbert’s fifth problem with Enflo’s geometric programme and modern developments beyond local compactness.
Presents original results by the authors, published here for the first time, including a universal formula for the critical path exponent and new quantitative square-root theorems.
Develops extensions to abelian and noncommutative metric groups using metric type, cube inequalities, and quantitative control of conjugation.
Uses explicit constructions, examples, and counterexamples to distinguish sharp regularity thresholds from the finer questions of critical endpoint behaviour.
Provides substantial preliminaries, detailed proofs of the main results, and a reader’s guide offering complementary routes through the material.
Includes a dedicated chapter of open problems, linking questions from Enflo’s original programme with new directions arising from the results, giving researchers and graduate students problems to tackle with the techniques developed in the book.