6.3 Combinatorial group theory in hyperbolic groups

Hyperbolic groups are finitely generated groups with Cayley graphs that satisfy specific geometric conditions. These groups have solvable word problems, linear Dehn functions, and solvable conjugacy problems, making them a rich area of study in geometric group theory.

Combinatorial group theory techniques are powerful tools for analyzing hyperbolic groups. From Dehn presentations to isoperimetric inequalities, these methods reveal key properties like finite asymptotic dimension, exponential growth rates, and regular geodesic languages in hyperbolic groups.

Combinatorial group theory for hyperbolic groups

Fundamental concepts and techniques

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• Hyperbolic groups represent finitely generated groups with Cayley graphs satisfying specific geometric conditions (δ-hyperbolic condition)
• Word problem in hyperbolic groups remains solvable through efficient algorithms based on Dehn's algorithm
• Linear Dehn functions measure the area of minimal van Kampen diagrams for null-homotopic words in hyperbolic groups
• Conjugacy problem in hyperbolic groups finds solutions through algorithms utilizing cyclic reduction
• Small cancellation theory (C'(1/6) and C'(1/4)-T(4) conditions) provides powerful analytical tools for hyperbolic groups

Advanced techniques and applications

• Rips construction and Stallings foldings adapt to study hyperbolic groups and their subgroups
• Gromov boundary of a hyperbolic group captures its asymptotic geometry through combinatorial analysis
• Dehn presentation property in finite presentations of hyperbolic groups enables efficient word problem solutions
• Isoperimetric inequality for hyperbolic groups linearly bounds the area of minimal van Kampen diagrams by boundary length
• Finite asymptotic dimension of hyperbolic groups derives from their finite presentations

Growth and language properties

• Exponential growth rate of hyperbolic groups computes from finite presentations using generating functions
• Novikov-Shubin invariants in finite presentations of hyperbolic groups provide information about spectral properties
• Language of geodesics in hyperbolic groups forms a regular language derived from finite presentations
• Hyperbolic groups contain finitely many conjugacy classes of finite subgroups determinable from finite presentations

Properties of finite presentations

Structural characteristics

• Finite presentations of hyperbolic groups exhibit the Dehn presentation property enabling efficient word problem solutions
• Isoperimetric inequality linearly bounds the area of minimal van Kampen diagrams by the length of their boundary
• Hyperbolic groups possess finite asymptotic dimension derivable from their finite presentations
• Exponential growth rate of hyperbolic groups computes from finite presentations using generating functions (power series)
• Novikov-Shubin invariants in finite presentations provide information about spectral properties of the group

Language and subgroup properties

• Geodesic language in hyperbolic groups forms a regular language derived from finite presentations
• Hyperbolic groups contain finitely many conjugacy classes of finite subgroups determinable from finite presentations
• Quasiconvex subgroups of hyperbolic groups remain finitely generated and hyperbolic, analyzable using combinatorial methods
• Intersection of two quasiconvex subgroups in a hyperbolic group maintains quasiconvexity, allowing systematic subgroup structure study
• Hyperbolic groups demonstrate the Howson property where the intersection of two finitely generated subgroups remains finitely generated

Structure of subgroups and quotients

Subgroup properties

• Quasiconvex subgroups of hyperbolic groups remain finitely generated and hyperbolic, analyzable through combinatorial methods
• Intersection of two quasiconvex subgroups in a hyperbolic group maintains quasiconvexity, enabling systematic subgroup structure study
• Hyperbolic groups exhibit the Howson property where the intersection of two finitely generated subgroups remains finitely generated
• Normal subgroups of infinite index in hyperbolic groups are either finite or possess uncountably many ends
• Virtual normalizer theorem states any infinite subgroup of a hyperbolic group has finite index in its normalizer

Quotient group characteristics

• Quotients of hyperbolic groups by finite normal subgroups remain hyperbolic, preserving numerous combinatorial properties
• Relatively hyperbolic groups, representing quotients of hyperbolic groups by parabolic subgroups, allow study using modified combinatorial techniques
• Rips complex of a hyperbolic group demonstrates contractibility, provable using combinatorial methods with implications for cohomological dimension
• Hyperbolic groups satisfy a linear isoperimetric inequality, provable using van Kampen diagrams and small cancellation theory
• Combination Theorem for hyperbolic groups enables construction of new hyperbolic groups from existing ones, involving analysis of amalgamated products and HNN extensions geometry

Combinatorial properties of hyperbolic groups

Geometric and topological properties

• Rips complex of a hyperbolic group exhibits contractibility, provable through combinatorial methods with implications for cohomological dimension
• Linear isoperimetric inequality satisfaction in hyperbolic groups proves using van Kampen diagrams and small cancellation theory
• Combination Theorem for hyperbolic groups allows new hyperbolic group construction from existing ones, analyzing amalgamated products and HNN extensions geometry
• Finite asymptotic dimension of hyperbolic groups proves using covering techniques and δ-hyperbolic spaces properties
• Morse Lemma for hyperbolic groups states quasigeodesics remain close to geodesics, provable using combinatorial arguments on the Cayley graph

Growth and subgroup structure

• Gromov's theorem on polynomial growth groups extends to hyperbolic groups, proving either polynomial or exponential growth
• Tits Alternative for hyperbolic groups states they either contain a free subgroup of rank 2 or are virtually cyclic, provable using combinatorial group theory techniques
• Normal subgroups of infinite index in hyperbolic groups are either finite or possess uncountably many ends
• Virtual normalizer theorem asserts any infinite subgroup of a hyperbolic group has finite index in its normalizer
• Hyperbolic groups contain finitely many conjugacy classes of finite subgroups, determinable from their finite presentations