S PECIAL W ORDS IN F REE G ROUPS S PECIAL W ORDS IN F REE G ROUPS Patrick Bishop, Cl�ment Gu�rin, Mary Leskovec, Vishal Mummareddy, and Tim Reid Mason Experimental Geometry Lab August 13, 2015
S PECIAL W ORDS IN F REE G ROUPS C ONTENTS I I NTRODUCTION Notation Words and Free Groups Special Words T HEORETICAL F ACTS Special in F 2 = Special in F r Higher Order Special Words Fricke Polynomial Reverse Pairs Inverse and Reverse Pairs are SL 2 Special SL 2 C special and GL 2 C special Are Equivalent Inverse and Reverse Pairs are not 3 Special The Positive Conjecture
S PECIAL W ORDS IN F REE G ROUPS C ONTENTS II C REATING A S PECIAL W ORD D ATABASE Computer Programs Statistics on the Special Words Generating All Words A NALYZING S PECIAL W ORDS Matrix Representation of Words F INDINGS Powers of Special Words Are Special All Special Words Must Have At Least 3 Instances Of A Letter The Image of a Special Pair by an Automorphism is Special H ISTOGRAMS F URTHER D IRECTIONS FOR R ESEARCH R EFERENCES
S PECIAL W ORDS IN F REE G ROUPS I NTRODUCTION N OTATION N OTATION U SED IN THE P RESENTATION ◮ SL n C Represents a n × n matrix whose elements are in the complex numbers and which has a determinant of 1. ◮ GL n C Represents a n × n matrix whose elements are in the complex numbers and which has a determinant not equal to 0. ◮ F r Represents a free group of r generators.
S PECIAL W ORDS IN F REE G ROUPS I NTRODUCTION W ORDS AND F REE G ROUPS F REE G ROUPS ◮ A group is a set whose elements are closed under some operation, have an identity value, have an inverse value, and are associative. ◮ A free group is a group whose elements, known as words, are strings of characters that are the group's generators. ◮ The operation, which the Free Group is closed under, is concatenation. ◮ For example, the Free Group of rank 2 is the group of all strings with the characters a , b , a − 1 , b − 1
S PECIAL W ORDS IN F REE G ROUPS I NTRODUCTION W ORDS AND F REE G ROUPS C YCLIC AND C ONJUGATE E QUIVALENCE ◮ A pair of words is cyclic if any of the two words can be rotated by taking the last element and moving it to the front to form the other word. ◮ A pair of words ( w 1 , w 2 ) is conjugate if they can be written as w 1 = pw 2 p − 1 for some word p. ◮ Cyclic equivalence ⇐ ⇒ Conjugate Equivalence ◮ This is because concatenation is the operation of the free group. wuw − 1 is cyclically equivalent to w − 1 wu , which is equal to u .
S PECIAL W ORDS IN F REE G ROUPS I NTRODUCTION W ORDS AND F REE G ROUPS W ORD MAP ◮ Take w ∈ F r w : ( SL n C ) r → SL n C ( A 1 , ..., A r ) �→ w ( A 1 , ..., A r ) In other words, it replaces every occurence of the generator a i of the free group by its corresponding matrix A i . The product of all of those gives one matrix.
S PECIAL W ORDS IN F REE G ROUPS I NTRODUCTION W ORDS AND F REE G ROUPS T RACE FUNCTION ◮ Take w ∈ F r Tr n ( w ) : ( SL n C ) r → C ( A 1 , ..., A r ) �→ Tr ( w ( A 1 , ..., A r ))
S PECIAL W ORDS IN F REE G ROUPS I NTRODUCTION W ORDS AND F REE G ROUPS T HE QUESTION We remark that conjugate words will always have the same trace function, the question is now, to which extent, the reverse of this implication is true.
S PECIAL W ORDS IN F REE G ROUPS I NTRODUCTION S PECIAL W ORDS S PECIAL W ORDS ◮ Two words (or more precisely their conjugacy classes), u , v ∈ F r , are n -special if u is not conjugate to v and their trace functions (in SL n C ) are equal. We say that they form a n -special pair. ◮ The set of special pairs for given n and r will be noted S n , r . ◮ A n -special word u ∈ F r is a word which is in some n -special pair.
S PECIAL W ORDS IN F REE G ROUPS I NTRODUCTION S PECIAL W ORDS A BUSE OF NOTATION For sake of simpli�cation we will talk about "words", however one should understand that this means "conjugacy classes of words". Most of our further algorithms require that the words are written under the following form : a α 1 b β 1 ... a α s b β s Actually one can verify that if a word is neither a k nor b l it is conjugate to a word of this form.
S PECIAL W ORDS IN F REE G ROUPS T HEORETICAL F ACTS S PECIAL IN F 2 = S PECIAL IN F r S PECIAL IN F 2 = S PECIAL IN F r ◮ This fact allows us to concentrate on words of 2 generators. ◮ S n , r ̸ = ∅ ⇐ ⇒ S n , 2 ̸ = ∅
S PECIAL W ORDS IN F REE G ROUPS T HEORETICAL F ACTS H IGHER O RDER S PECIAL W ORDS H IGHER O RDER S PECIALS A RE C ONTAINED I N S 2 ◮ The sets of higher order special words, if they exist, will have the following characteristic: S 1 ⊃ S 2 ⊃ S 3 ⊃ S 4 ⊃ S 5 ⊃ . . . S n Each set of special words is fully contained in the lower order sets. ◮ This is because any matrix A ∈ SL n − 1 matrix can be contained in an SL n matrix. P ROOF . Suppose a matrix A ∈ SL n − 1 . There exists a SL n matrix constructed as: ( A ) 0 0 1
S PECIAL W ORDS IN F REE G ROUPS T HEORETICAL F ACTS H IGHER O RDER S PECIAL W ORDS T HE I NTERSECTION OF H IGHER S PECIAL O RDERS IS E MPTY Another interesting fact is the following, all non-conjugate pairs won't have the same trace function in some SL n C , in other words : ∞ ∩ S n = ∅ n = 2
S PECIAL W ORDS IN F REE G ROUPS T HEORETICAL F ACTS F RICKE P OLYNOMIAL F RICKE P OLYNOMIAL AND F RICKE A LGORITHM ◮ For every w ∈ F 2 , there exists a unique polynomial P w ∈ Z [ X , Y , Z ] , called the Fricke Polynomial. For all A , B ∈ SL ( 2 , C ) , tr ( w ( A , B )) = P w ( tr ( A ) , tr ( B ) , tr ( AB )) . ◮ The polynomial is a result of using the trace relation ( tr ( AB ) = tr ( A ) tr ( B ) − tr ( AB − 1 ) ) to create a reduction algorithm that would take a word and break it down, and evaluate the Fricke polynomial of the word.
S PECIAL W ORDS IN F REE G ROUPS T HEORETICAL F ACTS F RICKE P OLYNOMIAL F RICKE P OLYNOMIAL E XAMPLE ◮ Example: tr ( a 4 bab ) = tr ( a 4 b ) tr ( ab ) − tr ( a 3 ) / tr ( a 2 ) tr ( a 2 b ) − tr ( b ) / tr ( a ) tr ( ab ) − tr ( b − 1 )
S PECIAL W ORDS IN F REE G ROUPS T HEORETICAL F ACTS R EVERSE P AIRS R EVERSE P AIRS ARE S PECIAL A reverse word is given by taking the elements of a word from last to �rst. r ( w )( a , b ) = w ( a − 1 , b − 1 ) − 1 Example: w ( a , b ) = ababbba r ( w ( a , b )) = ( a − 1 b − 1 a − 1 b − 1 b − 1 b − 1 a − 1 ) − 1 r ( w ( a , b )) = abbbaba
S PECIAL W ORDS IN F REE G ROUPS T HEORETICAL F ACTS I NVERSE AND R EVERSE P AIRS ARE SL 2 S PECIAL R EVERSE P AIRS ARE S PECIAL P ROOF . Suppose a word W ( a , b ) ∈ F 2 Tr ( W ( a , b )) = P W ( Tr ( a ) , Tr ( b ) , Tr ( a , b )) = P W ( Tr ( a − 1 ) , Tr ( b − 1 ) , Tr ( a − 1 b − 1 )) = Tr ( W ( a − 1 , b − 1 )) = Tr ( ← − − − − W ( a , b )) P W represents the Fricke Polynomial of the trace. If they are not equal, then they must also be special with each other.
S PECIAL W ORDS IN F REE G ROUPS T HEORETICAL F ACTS I NVERSE AND R EVERSE P AIRS ARE SL 2 S PECIAL I NVERSE P AIRS ARE S PECIAL Using the characteristic polynomial as applied to SL ( 2 , C ) : P ROOF . X 2 − tr ( X ) X + I = O X − 1 ( X 2 − tr ( X ) X + I ) = X − 1 O tr ( X ) − tr ( tr ( X ) I ) + tr ( X − 1 ) = tr ( O ) tr ( X ) − 2 tr ( X ) + tr ( X − 1 ) = 0 tr ( X ) = tr ( X − 1 )
S PECIAL W ORDS IN F REE G ROUPS T HEORETICAL F ACTS SL 2 C SPECIAL AND GL 2 C SPECIAL A RE E QUIVALENT SL S PECIAL I MPLIES GL S PECIAL P ROOF . Any matrix, A ∈ GL n C , can be transformed into an SL matrix by multiplying the matrix by the scalar: √ 1 Det ( A ) Since special words must have the same exponent, the scalars changing the GL matrices to SL matrices of special words will be equal. Since trace is linear, the traces will remain equal after the scalar multiplications. Therefore, if two words are special in SL 2 C , they are special in GL 2 C
S PECIAL W ORDS IN F REE G ROUPS T HEORETICAL F ACTS SL 2 C SPECIAL AND GL 2 C SPECIAL A RE E QUIVALENT GL S PECIAL I MPLIES SL S PECIAL P ROOF . Since SL 2 C ⊂ GL 2 C , if a words are special in GL 2 C then it is special in SL 2 C . Therefore, words are special in SL 2 C if and only if they words are special in GL 2 C .
1 and taking the trace we get : Multiplying by 1 2 2 2 S PECIAL W ORDS IN F REE G ROUPS T HEORETICAL F ACTS I NVERSE AND R EVERSE P AIRS ARE NOT 3 S PECIAL ( w , w − 1 ) IS NOT 3 - SPECIAL The proof involves two tools. First by standard analysis of the characteristic polynomial for A ∈ SL 3 C , we have : χ A ( X ) = X 3 − tr ( A ) X 2 + tr ( A − 1 ) X − 1 Using the Cayley-Hamilton theorem we get that : A 3 − tr ( A ) A 2 + tr ( A − 1 ) A = I 3
S PECIAL W ORDS IN F REE G ROUPS T HEORETICAL F ACTS I NVERSE AND R EVERSE P AIRS ARE NOT 3 S PECIAL ( w , w − 1 ) IS NOT 3 - SPECIAL The proof involves two tools. First by standard analysis of the characteristic polynomial for A ∈ SL 3 C , we have : χ A ( X ) = X 3 − tr ( A ) X 2 + tr ( A − 1 ) X − 1 Using the Cayley-Hamilton theorem we get that : A 3 − tr ( A ) A 2 + tr ( A − 1 ) A = I 3 Multiplying by A − 1 and taking the trace we get : 2 tr ( A − 1 ) = tr ( A ) 2 − tr ( A 2 )
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