Group Geodesic Growth Alex Bishop 29 June 2018 University of - PowerPoint PPT Presentation

Group Geodesic Growth Alex Bishop 29 June 2018 University of Technology Sydney

Definitions Let G be a group with symmetric finite generating set X . Recall that usual growth is defined as γ X ( n ) := # { g ∈ G : � g � X � n } 2

Definitions Let G be a group with symmetric finite generating set X . Recall that usual growth is defined as γ X ( n ) := # { g ∈ G : � g � X � n } Similarly, geodesic growth is defined as Γ X ( n ) := # { x 1 x 2 · · · x k ∈ X ∗ : � x 1 x 2 · · · x k � X = k � n } 2

Definitions Let G be a group with symmetric finite generating set X . Recall that usual growth is defined as γ X ( n ) := # { g ∈ G : � g � X � n } Similarly, geodesic growth is defined as Γ X ( n ) := # { x 1 x 2 · · · x k ∈ X ∗ : � x 1 x 2 · · · x k � X = k � n } Clearly, γ X ( n ) � Γ X ( n ) � | X | ( | X | − 1) n − 1 2

Example 1: Different Growth Classes . . . . . . . . . · · · · · · Presentation: · · · · · · Z 2 = � a, b | [ a, b ] � · · · · · · . . . . . . . . . Regular Growth: γ { a ± 1 ,b ± 1 } ( n ) = 2 n 2 + 2 n + 1 Geodesic Growth: Γ { a ± 1 ,b ± 1 } ( n ) = 2 n +3 − 4 n − 7 3

Example 2: Every Group has Exponential Geodesic Growth Presentation: Z = � a, b | a = b � Z = � z | −� · · · · · · · · · · · · Usual Growth: γ { z ± 1 } ( n ) = 2 n + 1 γ { a ± 1 ,b ± 1 } ( n ) = 2 n + 1 Geodesic Growth: Γ { a ± 1 ,b ± 1 } ( n ) = 2 n +2 − 3 Γ { z ± 1 } ( n ) = 2 n + 1 4

Example 3: 1 Different Geodesic Growth Rates Presentation: · · · · · · � � � � t 2 , [ a, t ] a, t � · · · · · · Geodesic Growth Rate: Γ { a ± 1 ,t } ( n ) = n 2 + 3 n (for n � 2 ) 1 Bridson, Burillo, M. Elder and Z. Šunić, ‘On groups whose geodesic growth is polynomial’, 2012. 5

Example 3: 1 Different Geodesic Growth Rates Presentation: · · · · · · � � � � t 2 , [ a, t ] a, t � · · · · · · Geodesic Growth Rate: Γ { a ± 1 ,t } ( n ) = n 2 + 3 n (for n � 2 ) Titze Transform: (where c = at ) · · · · · · � a 2 = c 2 , [ a, c ] � � � a, c � · · · · · · 1 Bridson, Burillo, M. Elder and Z. Šunić, ‘On groups whose geodesic growth is polynomial’, 2012. 5

Example 3: 1 Different Geodesic Growth Rates Presentation: · · · · · · � � � � t 2 , [ a, t ] a, t � · · · · · · Geodesic Growth Rate: Γ { a ± 1 ,t } ( n ) = n 2 + 3 n (for n � 2 ) Titze Transform: (where c = at ) · · · · · · � a 2 = c 2 , [ a, c ] � � � a, c � · · · · · · Geodesic Growth Rate: Γ { a ± 1 ,t } ( n ) = 2 n +1 − 1 1 Bridson, Burillo, M. Elder and Z. Šunić, ‘On groups whose geodesic growth is polynomial’, 2012. 5

Geodesic Growth of Z 2 Theorem (Proposition 10 2 ) Z 2 has exponential geodesic growth w.r.t. any finite generating set 2 Bridson, Burillo, M. Elder and Z. Šunić, ‘On groups whose geodesic growth is polynomial’, 2012. 6

Geodesic Growth of Z 2 Theorem (Proposition 10 2 ) Z 2 has exponential geodesic growth w.r.t. any finite generating set Theorem (Corollary 11 2 ) If G maps homomorphically onto Z 2 , then G has exponential geodesic growth w.r.t. any finite generating set 2 Bridson, Burillo, M. Elder and Z. Šunić, ‘On groups whose geodesic growth is polynomial’, 2012. 6

Example 4: 3 Virtually Z 2 Presentation: � [ a, b ] , t 2 , a t = b � � � a, b, t � Usual Growth: γ { a ± 1 ,b ± 1 ,t } ( n ) = 4 n 2 + 2 (for n � 2 ) Geodesic Growth: Γ { a ± 1 ,b ± 1 ,t } ( n ) = ( n + 1) · 2 n +2 − 2 n 2 − 6 n − 2 (for n � 5 ) 7 3 Bridson, Burillo, M. Elder and Z. Šunić, ‘On groups whose geodesic growth is polynomial’, 2012.

Example 4: 3 Virtually Z 2 Presentation: � [ a, b ] , t 2 , a t = b � � � a, b, t � Usual Growth: γ { a ± 1 ,b ± 1 ,t } ( n ) = 4 n 2 + 2 (for n � 2 ) Geodesic Growth: Γ { a ± 1 ,b ± 1 ,t } ( n ) = ( n + 1) · 2 n +2 − 2 n 2 − 6 n − 2 (for n � 5 ) Removing b : � � � a, a t � , t 2 � a, t � � 7 3 Bridson, Burillo, M. Elder and Z. Šunić, ‘On groups whose geodesic growth is polynomial’, 2012.

Example 4: 3 Virtually Z 2 Presentation: � [ a, b ] , t 2 , a t = b � � � a, b, t � Usual Growth: γ { a ± 1 ,b ± 1 ,t } ( n ) = 4 n 2 + 2 (for n � 2 ) Geodesic Growth: Γ { a ± 1 ,b ± 1 ,t } ( n ) = ( n + 1) · 2 n +2 − 2 n 2 − 6 n − 2 (for n � 5 ) Removing b : � � � a, a t � , t 2 � a, t � � Geodesic Growth: Γ { a ± 1 ,t ± 1 } ( n ) = 2 n 3 − 2 n + 18 (for n � 5 ) 3 7 3 Bridson, Burillo, M. Elder and Z. Šunić, ‘On groups whose geodesic growth is polynomial’, 2012.

Is there a group with intermediate geodesic growth?

Intermediate Usual Growth What about Grigorchuk’s group? 4 Elder, Gutierrez and Šunić, ‘Geodesics in the first Grigorchuk group’. 10

Intermediate Usual Growth What about Grigorchuk’s group? Consider the Schreier graphs b b b b b a a a a c c c c d c c d d d d d d 4 Elder, Gutierrez and Šunić, ‘Geodesics in the first Grigorchuk group’. 10

Intermediate Usual Growth What about Grigorchuk’s group? Consider the Schreier graphs b b b b b c b b b a a a a a a a a c c c c c c d d d c c c c d d d d d b b d d d d d 4 Elder, Gutierrez and Šunić, ‘Geodesics in the first Grigorchuk group’. 10

Intermediate Usual Growth What about Grigorchuk’s group? Consider the Schreier graphs b b b b b c b b b a a a a a a a a c c c c c c d d d c c c c d d d d d b b d d d d d The geodesic growth is exponential 4 4 Elder, Gutierrez and Šunić, ‘Geodesics in the first Grigorchuk group’. 10

Other Candidates for this Method Following this idea: (Brönnimann 5 ) Grigorchuk G ω : Gupta-Sidki p -groups: Square group: Spinal group: Gupta-Fabrykowski: 5 Brönnimann, ‘Geodesic growth of groups’, 2016. 11

Other Candidates for this Method Following this idea: (Brönnimann 5 ) Grigorchuk G ω : exponential Gupta-Sidki p -groups: Square group: Spinal group: Gupta-Fabrykowski: 5 Brönnimann, ‘Geodesic growth of groups’, 2016. 11

Other Candidates for this Method Following this idea: (Brönnimann 5 ) Grigorchuk G ω : exponential Gupta-Sidki p -groups: exponential Square group: Spinal group: Gupta-Fabrykowski: 5 Brönnimann, ‘Geodesic growth of groups’, 2016. 11

Other Candidates for this Method Following this idea: (Brönnimann 5 ) Grigorchuk G ω : exponential Gupta-Sidki p -groups: exponential Square group: exponential Spinal group: Gupta-Fabrykowski: 5 Brönnimann, ‘Geodesic growth of groups’, 2016. 11

Other Candidates for this Method Following this idea: (Brönnimann 5 ) Grigorchuk G ω : exponential Gupta-Sidki p -groups: exponential Square group: exponential Spinal group: exponential Gupta-Fabrykowski: 5 Brönnimann, ‘Geodesic growth of groups’, 2016. 11

Other Candidates for this Method Following this idea: (Brönnimann 5 ) Grigorchuk G ω : exponential Gupta-Sidki p -groups: exponential Square group: exponential Spinal group: exponential Gupta-Fabrykowski: . . . 5 Brönnimann, ‘Geodesic growth of groups’, 2016. 11

Other Candidates for this Method Following this idea: (Brönnimann 5 ) Grigorchuk G ω : exponential Gupta-Sidki p -groups: exponential Square group: exponential Spinal group: exponential Gupta-Fabrykowski: . . . technique doesn’t work 5 Brönnimann, ‘Geodesic growth of groups’, 2016. 11

Schreier Graph for Gupta-Fabrykowski 6 Bartholdi and Grigorchuk, ‘On the spectrum of Hecke type operators related to some fractal groups’, 2000. 12 7 Brönnimann, ‘Geodesic growth of groups’, 2016.

Gupta-Fabrykowski Let X = { 1 , 2 , 3 } and T 3 = X ω . Considering the wreath recursion Aut( T 3 ) = Aut( T 3 ) ≀ Sym( X ) then a = (1 , 1 , 1) · σ b = ( a, 1 , b ) · 1 where σ = (1 2 3) is a cyclic permutation of X . a 1 a 1 a 1 Then, together a and b generate the Gupta-Fabrykowski group. 13

Current Research: Experimental Mathematics I wrote a computer program for generating the geodesics of the Gupta-Fabrykowski group. 14

Current Research: Experimental Mathematics I wrote a computer program for generating the geodesics of the Gupta-Fabrykowski group. Why? 14

Current Research: Experimental Mathematics I wrote a computer program for generating the geodesics of the Gupta-Fabrykowski group. Why? • try to find an exponential growth sub-family 14

Current Research: Experimental Mathematics I wrote a computer program for generating the geodesics of the Gupta-Fabrykowski group. Why? • try to find an exponential growth sub-family • or some pattern to guide a proof of intermediate growth 14

Geodesic Patterns 1 1 1 length: 10 length: 25 length: 25 geodesics: 2 geodesics: 5 geodesics: 12 g 15 g g