Scientific Publishing THEORY OF BUILDINGS AND APPLICATIONS IN GEOMETRY AND TOPOLOGY

This book provides an accessible and application-oriented introduction to Tits buildings, a central geometric and combinatorial tool for understanding symmetry in algebraic and reductive groups. Symmetry plays a fundamental role across mathematics and science, and group theory offers the natural language for its study; Tits buildings, introduced by Jacques Tits, encode the rich structure of these groups in a highly structured geometric framework.The monograph emphasizes concrete examples, geometric intuition, and cross-disciplinary applications. Beginning with essential background in combinatorics, Chevalley groups, and reductive groups over local fields, the book develops the theory of spherical buildings, Euclidean (affine) buildings, and twin buildings in a gradual and transparent manner. The exposition is designed to lower the entry barrier to a subject often regarded as technically demanding.A distinctive feature of the book is its wide range of applications, illustrating how buildings arise naturally in geometry, topology, number theory, rigid analytic geometry, and group theory. Topics include geometric realizations of buildings, Euclidean buildings and their applications, ℝ-trees and ℝ-buildings, and connections with Kac–Moody groups. Elementary constructions and carefully chosen examples clarify why buildings are powerful tools for studying symmetry and group actions on geometric spaces.By offering a unified, intuitive, and example-driven treatment, this book serves as an ideal entry point for graduate students and researchers seeking to understand what Tits buildings are, why they matter, and how they are used in modern mathematics.