Geometry of the conjugacy problem Andrew Sale Vanderbilt University May 14, 2015 Andrew Sale Geometry of the conjugacy problem
For a group G , we define two functions: Andrew Sale Geometry of the conjugacy problem
For a group G , we define two functions: 1 Conjugacy length function Bounds length of short conjugators. Andrew Sale Geometry of the conjugacy problem
For a group G , we define two functions: 1 Conjugacy length function Bounds length of short conjugators. 2 Permutation conjugacy length function Inspired by fast solutions to the conjugacy problem in hyerbolic and relatively hyperbolic groups (Bridson–Howie, Epstein–Holt, Bumagin). Andrew Sale Geometry of the conjugacy problem
Conjugacy Length Function G group with length function |·| : G → [0 , ∞ ) (e.g. word length if finitely generated). Andrew Sale Geometry of the conjugacy problem
Conjugacy Length Function G group with length function |·| : G → [0 , ∞ ) (e.g. word length if finitely generated). Definition (Conjugacy length function) CLF G : [0 , ∞ ) → [0 , ∞ ) minimal function satisfying: Andrew Sale Geometry of the conjugacy problem
Conjugacy Length Function G group with length function |·| : G → [0 , ∞ ) (e.g. word length if finitely generated). Definition (Conjugacy length function) CLF G : [0 , ∞ ) → [0 , ∞ ) minimal function satisfying: For x ≥ 0 , u, v ∈ G such that | u | + | v | ≤ x , then Andrew Sale Geometry of the conjugacy problem
Conjugacy Length Function G group with length function |·| : G → [0 , ∞ ) (e.g. word length if finitely generated). Definition (Conjugacy length function) CLF G : [0 , ∞ ) → [0 , ∞ ) minimal function satisfying: For x ≥ 0 , u, v ∈ G such that | u | + | v | ≤ x , then ∃ g ∈ G such that (i) gug − 1 = v and u is conjugate to v ⇐ ⇒ Andrew Sale Geometry of the conjugacy problem
Conjugacy Length Function G group with length function |·| : G → [0 , ∞ ) (e.g. word length if finitely generated). Definition (Conjugacy length function) CLF G : [0 , ∞ ) → [0 , ∞ ) minimal function satisfying: For x ≥ 0 , u, v ∈ G such that | u | + | v | ≤ x , then ∃ g ∈ G such that (i) gug − 1 = v and u is conjugate to v ⇐ ⇒ (ii) | g | ≤ CLF G ( x ) . Andrew Sale Geometry of the conjugacy problem
Conjugacy Length Function G group with length function |·| : G → [0 , ∞ ) (e.g. word length if finitely generated). Definition (Conjugacy length function) CLF G : [0 , ∞ ) → [0 , ∞ ) minimal function satisfying: For x ≥ 0 , u, v ∈ G such that | u | + | v | ≤ x , then ∃ g ∈ G such that (i) gug − 1 = v and u is conjugate to v ⇐ ⇒ (ii) | g | ≤ CLF G ( x ) . Lemma Γ finitely generated with solvable WP, |·| word length. Then: Conjugacy problem is solvable ⇐ ⇒ CLF Γ is recursive. Andrew Sale Geometry of the conjugacy problem
Example: free groups F free group, finite generating set X . e.g. u = aabbbaba − 1 u, v reduced words on X ∪ X − 1 . v = babababba − 1 b − 1 Andrew Sale Geometry of the conjugacy problem
Example: free groups F free group, finite generating set X . e.g. u = aabbbaba − 1 u, v reduced words on X ∪ X − 1 . v = babababba − 1 b − 1 Algorithm to solve conjugacy problem Andrew Sale Geometry of the conjugacy problem
Example: free groups F free group, finite generating set X . e.g. u = aabbbaba − 1 u, v reduced words on X ∪ X − 1 . v = babababba − 1 b − 1 Algorithm to solve conjugacy problem (i) u ′ = a − 1 ua = ab 3 ab (i) Cyclically reduce u, v to u ′ , v ′ , v ′ = ( ba ) − 1 vba = babab 2 Andrew Sale Geometry of the conjugacy problem
Example: free groups F free group, finite generating set X . u = aabbbaba − 1 e.g. u, v reduced words on X ∪ X − 1 . v = babababba − 1 b − 1 Algorithm to solve conjugacy problem (i) u ′ = a − 1 ua = ab 3 ab (i) Cyclically reduce u, v to u ′ , v ′ , v ′ = ( ba ) − 1 vba = babab 2 (ii) Cyclically conjugate u ′ to v ′ . (ii) v ′ = babu ′ ( bab ) − 1 Andrew Sale Geometry of the conjugacy problem
Example: free groups F free group, finite generating set X . u = aabbbaba − 1 e.g. u, v reduced words on X ∪ X − 1 . v = babababba − 1 b − 1 Algorithm to solve conjugacy problem (i) u ′ = a − 1 ua = ab 3 ab (i) Cyclically reduce u, v to u ′ , v ′ , v ′ = ( ba ) − 1 vba = babab 2 (ii) Cyclically conjugate u ′ to v ′ . (ii) v ′ = babu ′ ( bab ) − 1 The conjugator will be a product g = bababa − 1 of subwords of u and v . Hence v = gug − 1 CLF F ( x ) ≤ x. Andrew Sale Geometry of the conjugacy problem
State of the art Known results include: Class of groups CLF( x ) Hyperbolic groups linear Lysenok CAT(0) & biautomatic groups � exp( x ) Bridson–Haefliger RAAGs & special subgroups linear Crisp–Godelle–Wiest Mapping class groups linear Masur–Minsky; Behrstock–Drut ¸u; J. Tao. 2-Step Nilpotent quadratic Ji–Ogle–Ramsey � x 2 π 1 ( M ) , M prime 3 –manifold Behrstock–Drut ¸u, S � x 3 Free solvable groups S Plus: wreath products (S), group extensions (S), relatively hyperbolic groups (Ji–Ogle–Ramsey, Z. O’Conner, Bumagin). Andrew Sale Geometry of the conjugacy problem
Permutation conjugacy length function, j/w Y. Antol´ ın. G group, X (finite) generating set, |·| word length. Definition (Permutation conjugacy length function) PCL G,X : N → N minimal function satisfying: Andrew Sale Geometry of the conjugacy problem
Permutation conjugacy length function, j/w Y. Antol´ ın. G group, X (finite) generating set, |·| word length. Definition (Permutation conjugacy length function) PCL G,X : N → N minimal function satisfying: For geodesic words u, v on X such that | u | + | v | ≤ n , then u, v represent conjugate elements of G iff Andrew Sale Geometry of the conjugacy problem
Permutation conjugacy length function, j/w Y. Antol´ ın. G group, X (finite) generating set, |·| word length. Definition (Permutation conjugacy length function) PCL G,X : N → N minimal function satisfying: For geodesic words u, v on X such that | u | + | v | ≤ n , then u, v represent conjugate elements of G iff ∃ cyclic permutations u ′ , v ′ of u, v and g ∈ G such that (i) gu ′ g − 1 = v ′ and (ii) | g | ≤ PCL G,X ( n ) . Andrew Sale Geometry of the conjugacy problem
Permutation conjugacy length function, j/w Y. Antol´ ın. G group, X (finite) generating set, |·| word length. Definition (Permutation conjugacy length function) PCL G,X : N → N minimal function satisfying: For geodesic words u, v on X such that | u | + | v | ≤ n , then u, v represent conjugate elements of G iff ∃ cyclic permutations u ′ , v ′ of u, v and g ∈ G such that (i) gu ′ g − 1 = v ′ and (ii) | g | ≤ PCL G,X ( n ) . e.g. For a free group PCL = 0 . Andrew Sale Geometry of the conjugacy problem
Sublinear PCL Relationship to CLF : PCL G,X ( n ) ≤ CLF G ( n ) ≤ PCL G,X ( n ) + n. v w uw = wv CLF G ( n ) PCL G,X ( n ) u u 1 Andrew Sale Geometry of the conjugacy problem
Sublinear PCL Relationship to CLF : PCL G,X ( n ) ≤ CLF G ( n ) ≤ PCL G,X ( n ) + n. v uw = wv w CLF( n ) PCL( n ) u u 1 If PCL G,X ( n ) ≤ K for all n , then conjugacy problem is almost as fast as word problem: (on input geodesic words). Apply the word problem n 2 times, on words of length n + 2 K , where n is the sum of the length of the input words. Andrew Sale Geometry of the conjugacy problem
Relatively hyperbolic groups Theorem (Antol´ ın–S ’15) Let G be hyperbolic relative to a finite collection of subgroups { H ω } ω ∈ Ω . There exists a finite generating set X such that � X ∩ H ω � = H ω and � � PCL G,X ( n ) � max PCL H ω ,X ∩ H ω ( n ) . ω ∈ Ω Andrew Sale Geometry of the conjugacy problem
Relatively hyperbolic groups Theorem (Antol´ ın–S ’15) Let G be hyperbolic relative to a finite collection of subgroups { H ω } ω ∈ Ω . There exists a finite generating set X such that � X ∩ H ω � = H ω and � � PCL G,X ( n ) � max PCL H ω ,X ∩ H ω ( n ) . ω ∈ Ω In particular, hyperbolic groups and groups that are hyperbolic relative to abelian groups will all have PCL bounded by a constant. Andrew Sale Geometry of the conjugacy problem
Consequences of constant PCL Suppose PCL G,X ( n ) ≤ K . 1 Potentially fast algorithm to solve the conjugacy problem. Andrew Sale Geometry of the conjugacy problem
Consequences of constant PCL Suppose PCL G,X ( n ) ≤ K . 1 Potentially fast algorithm to solve the conjugacy problem. 2 Exponential conjugacy growth rate controlled by exponential growth rate. Andrew Sale Geometry of the conjugacy problem
Consequences of constant PCL Suppose PCL G,X ( n ) ≤ K . 1 Potentially fast algorithm to solve the conjugacy problem. 2 Exponential conjugacy growth rate controlled by exponential growth rate. 3 (Ciobanu-Hermiller-Holt-Rees) ConjGeo ( G, X ) is a regular language whenever either Geo ( G, X ) has a biautomatic structure, ( G, X ) has falsification by fellow traveller property. Andrew Sale Geometry of the conjugacy problem
PCL G,X ( n ) < K for hyperbolic groups G hyperbolic. Take u, v geodesic words, conjugate in G . Cyclic permutations u ′ = u 2 u 1 , v ′ = v 2 v 1 and v ′ w = wu ′ with | w | minimal. Let w i be prefix of w . Andrew Sale Geometry of the conjugacy problem
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