12.1 The word and conjugacy problems

The word and conjugacy problems are fundamental in group theory, tackling the question of element equality and conjugacy. These problems highlight the complexity of group structures and their computational challenges, connecting abstract algebra with algorithmic thinking.

Solving these problems has far-reaching implications in mathematics and computer science. From cryptography to topology, understanding when and how these problems can be solved provides crucial insights into group properties and shapes our approach to computational group theory.

The Word Problem in Group Theory

Definition and Significance

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• Word problem determines if two words represent the same element in a group presentation
• Group presentation comprises generators and defining relations among them
• Asks whether an algorithm exists to decide if two words in generators are equivalent under given relations
• Connects to fundamental question of equality between group elements
• Undecidable in general (proven by Pyotr Novikov in 1955 and William Boone in 1958)
• Solvability or unsolvability provides insight into group's structure and complexity
• Relates to other decision problems (isomorphism problem, conjugacy problem)

Mathematical Formulation

• Given: Group $G$ with presentation $\langle X | R \rangle$
• $X$ represents generators, $R$ represents relations
• Two words $w_1$ and $w_2$ in $X^*$ (free monoid on $X$)
• Problem asks if $w_1 =_G w_2$ (equality in $G$)
• Equivalent to asking if $w_1w_2^{-1} =_G 1$ (trivial element in $G$)
• Example: In group $\langle a,b | a^2=b^3=1 \rangle$, determine if $abab = baba$
  • Rewrite as $abab(baba)^{-1} = ababab^{-1}a^{-1}b^{-1} =_G 1$
  • Apply relations: $a^2=1$, $b^3=1$
  • Simplify: $abab^2a = 1$

Implications and Applications

• Solvability impacts computational group theory and geometric group theory
• Unsolvable word problem indicates group has complex structure
• Solvable word problem allows for effective computations within the group
• Applications in cryptography (group-based cryptosystems)
• Used in studying group actions on geometric spaces
• Influences design of computer algebra systems for group computations
• Example: Hyperbolic groups (word problem solvable in linear time)
  • Dehn's algorithm for solving word problem
  • Applications in 3-manifold topology

Conjugacy Problem vs Word Problem

Definitions and Relationships

• Conjugacy problem determines if two group elements are conjugate
• Elements $a$ and $b$ in group $G$ are conjugate if $\exists g \in G: g^{-1}ag = b$
• Generalizes word problem (equality implies conjugacy)
• Solvable conjugacy problem implies solvable word problem
• Converse not true (groups with solvable word problem, unsolvable conjugacy problem exist)
• Example: In symmetric group $S_n$, conjugacy related to cycle structure
  • $(1 2 3)$ and $(2 3 4)$ are conjugate in $S_4$
  • $(1 2)(3 4)$ and $(1 2 3)$ are not conjugate in $S_4$

Significance and Applications

• Important in cryptography (designing conjugacy-based cryptosystems)
• Undecidable in general for finitely presented groups
• Provides deeper insight into group structure than word problem alone
• Used in studying automorphisms and outer automorphisms of groups
• Applications in geometric group theory and topology
• Example: Braid groups
  • Both word and conjugacy problems solvable
  • Conjugacy problem crucial for braid-based cryptography

Complexity Comparison

• Conjugacy problem often more difficult than word problem
• Solving techniques may differ significantly between the two problems
• Time complexity of conjugacy problem typically higher than word problem
• Example: Free groups
  • Word problem solvable in linear time
  • Conjugacy problem solvable in linear time for cyclically reduced words
• Example: Braid groups
  • Word problem solvable in quadratic time
  • Best known conjugacy algorithms have exponential time complexity

Solvability of Word and Conjugacy Problems

Solvable Cases

• Finite groups (exhaustive checking possible)
• Free groups (normal form algorithms)
• Hyperbolic groups (Dehn's algorithm)
• One-relator groups (Magnus' theorem for word problem)
• Braid groups (Garside normal form)
• Automatic groups (includes hyperbolic groups and braid groups)
• Example: Free group on two generators $F_2 = \langle a, b \rangle$
  • Word problem: Freely reduce words (cancel $aa^{-1}$, $bb^{-1}$, etc.)
  • Conjugacy problem: Cyclically reduce words, then check if cyclic permutations

Unsolvable Cases

• Exist groups with unsolvable word problem
• Groups with solvable word problem but unsolvable conjugacy problem
• Certain wreath products (Charles F. Miller III's construction)
• Some finitely presented groups derived from Turing machines
• Example: $F_2 \times F_2$ (direct product of two free groups)
  • Word problem solvable
  • Conjugacy problem unsolvable (Mikhailova's construction)

Partial Results and Special Cases

• Nilpotent groups (word problem solvable in polynomial time)
• Polycyclic groups (word problem solvable, conjugacy problem open for some cases)
• Metabelian groups (word problem solvable, conjugacy problem generally unsolvable)
• CAT(0) groups (word problem solvable, conjugacy problem open in general)
• Example: Nilpotent group of class 2
  • Word problem solvable in cubic time
  • Conjugacy problem solvable but potentially more complex

Algorithms for Word and Conjugacy Problems

Word Problem Algorithms

• Free groups: Free reduction algorithm
  • Cancel adjacent inverse generator pairs
  • Example: $abba^{-1}b^{-1}a \rightarrow aba$
• Abelian groups: Count occurrences of generators
  • Compare resulting vectors
  • Example: In $\mathbb{Z}^2$, $a^2b^3a^{-1}b = a^1b^4$
• Small cancellation groups: Dehn's algorithm
  • Replace subwords with shorter equivalent words
  • Example: In $\langle a,b | abab^{-1}a^{-1}b^{-1} = 1 \rangle$, replace $abab^{-1}$ with $ba$
• Hyperbolic groups: Linear-time Dehn's algorithm
• Nilpotent groups: Collection process
  • Polynomial time in word length (degree depends on nilpotency class)
• Automatic groups: Use finite state automata
  • Quadratic time algorithm based on fellow traveler property

Conjugacy Problem Algorithms

• Braid groups: Garside normal form algorithm
  • Works for both word and conjugacy problems
  • Example: In braid group $B_3$, $\sigma_1\sigma_2\sigma_1 \sim \sigma_2\sigma_1\sigma_2$
• Free groups: Cyclic reduction and permutation
  • Cyclically reduce words, then check if one is a cyclic permutation of the other
  • Example: In $F_2$, $ababa \sim babab$
• Hyperbolic groups: Bounded geodesic image algorithm
  • Based on properties of quasi-geodesics in hyperbolic space
• Polycyclic groups: Collection and comparison of conjugates
  • Efficient for groups with small Hirsch length
• CAT(0) groups: Flat torus theorem for abelian subgroups
  • Reduces problem to studying centralizers