Decimation Structure of the Spectra ofSelf-Similar Groups

Brett Hungar, Madison Phelps, and Jonathan Wheeler

University of Connecticut

July 23, 2019

1

Outline

1. Self-Similar groups

2. The Hanoi Towers Group

3. Spectral Similarity

4. The Grigorchuk Method

5. Future Research

Definition (Spectrum [Str06])The spectrum of a finite dimensional operator is the set ofeigenvalues of that operator, denoted σpxq.

2

Outline

1. Self-Similar groups

2. The Hanoi Towers Group

3. Spectral Similarity

4. The Grigorchuk Method

5. Future Research

Definition (Spectrum [Str06])The spectrum of a finite dimensional operator is the set ofeigenvalues of that operator, denoted σpxq.

2

Self Similar Groups

Definition (Restriction of an automorphism [KSW12])Let X˚ be an infinite tree over an alphabet X˚ Let g be a treeautomorphism of X˚, and let x P X. Then xX˚ is naturallyisomorphic to X˚. Since g is an automorphism, restricting g toxX˚ gives an automorphism on xX˚. In turn, this givesanother automorphism on X˚, which we denote g|x.

g g|1H

1 0

11 10 00 01

1

11 10

110 111 100 101

3

Self Similar Groups

Definition (Restriction of an automorphism [KSW12])Let X˚ be an infinite tree over an alphabet X˚ Let g be a treeautomorphism of X˚, and let x P X. Then xX˚ is naturallyisomorphic to X˚. Since g is an automorphism, restricting g toxX˚ gives an automorphism on xX˚. In turn, this givesanother automorphism on X˚, which we denote g|x.

g g|1H

1 0

11 10 00 01

1

11 10

110 111 100 101

3

Self Similar Groups

Definition (Self similar groups [KSW12; Nek05])Let G be a group that acts faithfully on X˚. Then G is a selfsimilar group if, for all g P G and x P X there is a h P G andy P X such that gpxwq “ yhpwq. Equivalently, for all g P G andx P X, we have g|x P G.

4

Example: The Hanoi Towers Group

The Hanoi Towers game is played on k pegs with n disks. Thedisks are labeled 1 through n. Players can move disks from oneadjacent peg to another, but cannot place a larger disk on topof a smaller disk. [GS08]

Let X “ t0, 1, . . . , k ´ 1u. Then we can uniquely represent eachgame state as a word in Xn.

1200 112

5

Example: The Hanoi Towers Group

The Hanoi Towers game is played on k pegs with n disks. Thedisks are labeled 1 through n. Players can move disks from oneadjacent peg to another, but cannot place a larger disk on topof a smaller disk. [GS08]

Let X “ t0, 1, . . . , k ´ 1u. Then we can uniquely represent eachgame state as a word in Xn.

1200 112

5

Example: The Hanoi Towers Group

Definition (The Hanoi Towers group [GS08])The Hanoi Towers group H is generated by a, b, c, where arepresents switching a disk between pegs 0 and 1, b representsswitching a disk between 0 and 2, and c represents switchingbetween 1 and 2.

The Hanoi Towers group is defined in terms of its action on theset of game states of the Hanoi Towers game. Note that thisaction is always unique due to the rules of the game

6

Example: The Hanoi Towers Group

Definition (The Hanoi Towers group [GS08])The Hanoi Towers group H is generated by a, b, c, where arepresents switching a disk between pegs 0 and 1, b representsswitching a disk between 0 and 2, and c represents switchingbetween 1 and 2.

The Hanoi Towers group is defined in terms of its action on theset of game states of the Hanoi Towers game. Note that thisaction is always unique due to the rules of the game

6

Example: The Hanoi Towers Group

aÝÝÑ

0200

bÝÝÑ

1200 1000

cÝÝÑ

2200

7

Example: The Hanoi Towers Group

Since we can represent each game state as a word over X “

t0, 1, 2u, H also acts on X˚.

H

1 0 2

10 11 12 00 01 02 21 20 22

We can specify a group element by its permutation on the firstlevel and what it restricts to on lower levels. For Hanoi:

a “ p0 1qpid, id,aq

b “ p0 2qpid, b, idq

c “ p1 2qpc, id, idq

8

Example: The Hanoi Towers Group

Since we can represent each game state as a word over X “

t0, 1, 2u, H also acts on X˚.

H

1 0 2

10 11 12 00 01 02 21 20 22

We can specify a group element by its permutation on the firstlevel and what it restricts to on lower levels. For Hanoi:

a “ p0 1qpid, id,aq

b “ p0 2qpid, b, idq

c “ p1 2qpc, id, idq

8

Schreier graphs

Definition (Schreier graphs [GS08])Let G be a self similar group acting on X˚. Let S be a finitegenerating set of G. Then the nth Schreier graph of G withrespect to S is a directed graph with vertices Xn and an edgefrom x to y if there is a g P S such that gx “ y.

9

Spectral Similarity

Definition (Spectral similarity [MT03])We call an operator H spectrally similar to an operator H0 ifwe have complex functions ϕ0 and ϕ1 such that, when defined,

U˚pH ´ zq´1U “ pϕ0pzqH0 ´ ϕ1pzqq´1

Spectral similarity is a way to relate the spectra of two differentoperators. If we take

Rpzq “ϕ1pzq

ϕ0pzq

we see that z P σpHq means Rpzq P σpH0q.In our context, this means relating operators on graph approx-imations of fractals or Schreier graphs of self-similar groups.

10

Spectral Similarity

Definition (Spectral similarity [MT03])We call an operator H spectrally similar to an operator H0 ifwe have complex functions ϕ0 and ϕ1 such that, when defined,

U˚pH ´ zq´1U “ pϕ0pzqH0 ´ ϕ1pzqq´1

Spectral similarity is a way to relate the spectra of two differentoperators. If we take

Rpzq “ϕ1pzq

ϕ0pzq

we see that z P σpHq means Rpzq P σpH0q.In our context, this means relating operators on graph approx-imations of fractals or Schreier graphs of self-similar groups.

10

Example of Spectral Similarity

The probability Laplacian of n-level approximation of the Sier-pinski gasket is spectrally similar to the (n ´ 1)-level approxi-mation with Rpzq “ zp5´ 4zq.

σpSG1q “ t32 ,

32 , 0u σpSG2q “ t

32 ,

32 ,

32 ,

34 ,

34 , 0u

Plugging the values from σpSG2q into R gives you the valuesof σpSG1q. This gives a clear representation of a simple case ofspectral similarity.

11

Example of Spectral Similarity

The probability Laplacian of n-level approximation of the Sier-pinski gasket is spectrally similar to the (n ´ 1)-level approxi-mation with Rpzq “ zp5´ 4zq.

σpSG1q “ t32 ,

32 , 0u σpSG2q “ t

32 ,

32 ,

32 ,

34 ,

34 , 0u

Plugging the values from σpSG2q into R gives you the valuesof σpSG1q. This gives a clear representation of a simple case ofspectral similarity.

11

Example of Spectral Similarity

The probability Laplacian of n-level approximation of the Sier-pinski gasket is spectrally similar to the (n ´ 1)-level approxi-mation with Rpzq “ zp5´ 4zq.

σpSG1q “ t32 ,

32 , 0u σpSG2q “ t

32 ,

32 ,

32 ,

34 ,

34 , 0u

Plugging the values from σpSG2q into R gives you the valuesof σpSG1q. This gives a clear representation of a simple case ofspectral similarity.

11

The Grigorchuk Method: Group Representation

The action of H on level n induces a permutational 3n-dimensional representation,

ρn : H Ñ GLp3n,Cq of H

Denote ρnpaq “ an, ρnpbq “ bn, and ρnpcq “ cn

We represent the generators with permutational matrices recur-sively by,

a0 “ b0 “ c0 “ r1s

an`1 “

»

–

0 1 01 0 00 0 an

fi

fl bn`1 “

»

–

0 0 10 bn 01 0 0

fi

fl cn`1 “

»

–

cn 0 00 0 10 1 0

fi

fl

where each block is of size 3n ˆ 3n for n ě 0.

[GS08]

12

The Grigorchuk Method: Group Representation

The action of H on level n induces a permutational 3n-dimensional representation,

ρn : H Ñ GLp3n,Cq of H

Denote ρnpaq “ an, ρnpbq “ bn, and ρnpcq “ cn

We represent the generators with permutational matrices recur-sively by,

a0 “ b0 “ c0 “ r1s

an`1 “

»

–

0 1 01 0 00 0 an

fi

fl bn`1 “

»

–

0 0 10 bn 01 0 0

fi

fl cn`1 “

»

–

cn 0 00 0 10 1 0

fi

fl

where each block is of size 3n ˆ 3n for n ě 0.

[GS08]

12

The Grigorchuk Method: Group Representation

The action of H on level n induces a permutational 3n-dimensional representation,

ρn : H Ñ GLp3n,Cq of H

Denote ρnpaq “ an, ρnpbq “ bn, and ρnpcq “ cn

We represent the generators with permutational matrices recur-sively by,

a0 “ b0 “ c0 “ r1s

an`1 “

»

–

0 1 01 0 00 0 an

fi

fl bn`1 “

»

–

0 0 10 bn 01 0 0

fi

fl cn`1 “

»

–

cn 0 00 0 10 1 0

fi

fl

where each block is of size 3n ˆ 3n for n ě 0.

[GS08]

12

The Grigorchuk Method: Group Representation

The action of H on level n induces a permutational 3n-dimensional representation,

ρn : H Ñ GLp3n,Cq of H

Denote ρnpaq “ an, ρnpbq “ bn, and ρnpcq “ cn

We represent the generators with permutational matrices recur-sively by,

a0 “ b0 “ c0 “ r1s

an`1 “

»

–

0 1 01 0 00 0 an

fi

fl bn`1 “

»

–

0 0 10 bn 01 0 0

fi

fl cn`1 “

»

–

cn 0 00 0 10 1 0

fi

fl

where each block is of size 3n ˆ 3n for n ě 0. [GS08]

12

The Grigorchuk Method: The Adjacency Matrix

The Grigorchuk method uses the representation of each gener-ator to create an adjacency matrix that describes the action oneach sequence Schreier graphs.

For the Hanoi Towers group, the adjacency matrix is

∆n “ an ` bn ` cn

Equivalently,

∆n “

»

–

cn 1 11 bn 11 1 an

fi

fl

13

The Grigorchuk Method: The Adjacency Matrix

The Grigorchuk method uses the representation of each gener-ator to create an adjacency matrix that describes the action oneach sequence Schreier graphs.

For the Hanoi Towers group, the adjacency matrix is

∆n “ an ` bn ` cn

Equivalently,

∆n “

»

–

cn 1 11 bn 11 1 an

fi

fl

13

The Grigorchuk Method: The Adjacency Matrix

The Grigorchuk method uses the representation of each gener-ator to create an adjacency matrix that describes the action oneach sequence Schreier graphs.

For the Hanoi Towers group, the adjacency matrix is

∆n “ an ` bn ` cn

Equivalently,

∆n “

»

–

cn 1 11 bn 11 1 an

fi

fl

13

The Grigorchuk Method: The Adjacency Matrix

The adjacency for level 2 of the Hanoi Towers group;

∆2 “

»

—

—

—

—

—

—

—

—

—

—

—

–

c 0 0 1 0 0 1 0 00 0 1 0 1 0 0 1 00 1 0 0 0 1 0 0 11 0 0 0 0 1 1 0 00 1 0 0 b 0 0 1 00 0 1 1 0 0 0 0 11 0 0 1 0 0 0 1 00 1 0 0 1 0 1 0 00 0 1 0 0 1 0 0 a

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

14

Finding the spectrum

We now introduce a new variable x, [GNS15]. Define∆npxq “ ∆n ´ xI, and the spectrum of ∆n is all x where|∆npxq| “ 0.For example, level 2, we have

»

—

—

—

—

—

—

—

—

—

—

—

–

c ´ x 0 0 1 0 0 1 0 00 ´x 1 0 1 0 0 1 00 1 ´x 0 0 1 0 0 11 0 0 ´x 0 1 1 0 00 1 0 0 b ´ x 0 0 1 00 0 1 1 0 ´x 0 0 11 0 0 1 0 0 ´x 1 00 1 0 0 1 0 1 ´x 00 0 1 0 0 1 0 0 a ´ x

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

15

Moving to a two-dimensional dynamic system

Our goal is to write the determinant of ∆npxq in terms of thedeterminant of ∆n´1pxq. Unfortunately, the linear algebra doesnot allow us to do this (at least, not yet). [GS08]

Instead, we take ∆npx, yq “ ∆npxq ´ p1´ yqdn, where

dn`1 “

»

–

0 1n 1n1n 0 1n1n 1n 0

fi

fl

For example:

∆1px, yq “

»

–

1´ x y yy 1´ x yy y 1´ x

fi

fl

16

Moving to a two-dimensional dynamic system

Our goal is to write the determinant of ∆npxq in terms of thedeterminant of ∆n´1pxq. Unfortunately, the linear algebra doesnot allow us to do this (at least, not yet). [GS08]

Instead, we take ∆npx, yq “ ∆npxq ´ p1´ yqdn, where

dn`1 “

»

–

0 1n 1n1n 0 1n1n 1n 0

fi

fl

For example:

∆1px, yq “

»

–

1´ x y yy 1´ x yy y 1´ x

fi

fl

16

Auxiliary spectrum

Now we can express the determinant of ∆npx, yq in a recursiveformula

|∆npx, yq| “ Pnpx, yq|∆n´1pF px, yqq|

where F px, yq “ px1, y1q is a rational function in x and y andPnpx, yq is a polynomial. [GS08]

The zeroes of ∆npx, yq give us the auxiliary spectrum

Images of the auxiliary spectrum of levels 1, 2, and 3 [GS06].

17

Auxiliary spectrum

Now we can express the determinant of ∆npx, yq in a recursiveformula

|∆npx, yq| “ Pnpx, yq|∆n´1pF px, yqq|

where F px, yq “ px1, y1q is a rational function in x and y andPnpx, yq is a polynomial. [GS08]

The zeroes of ∆npx, yq give us the auxiliary spectrum

Images of the auxiliary spectrum of levels 1, 2, and 3 [GS06].

17

Converting from the auxiliary spectrum to the Schreierspectrum

Note that F takes points in the auxiliary spectrum to points inthe auxiliary spectrum. Our new goal is to map the auxiliaryspectrum to the Schreier spectrum.

R2 R2

R R

F

Ψ Ψ

f

Formally, we want to find Ψ and f that make this diagram commute.

As it turns out, Ψ and f do exist, and fpxq “ x2´x´3. [GS08]If x P σp∆npxqq, then fpxq P σp∆n´1pxqq. We can use this factto find the spectrum for any level.

18

Converting from the auxiliary spectrum to the Schreierspectrum

Note that F takes points in the auxiliary spectrum to points inthe auxiliary spectrum. Our new goal is to map the auxiliaryspectrum to the Schreier spectrum.

R2 R2

R R

F

Ψ Ψ

f

Formally, we want to find Ψ and f that make this diagram commute.

As it turns out, Ψ and f do exist, and fpxq “ x2´x´3. [GS08]If x P σp∆npxqq, then fpxq P σp∆n´1pxqq. We can use this factto find the spectrum for any level.

18

Converting from the auxiliary spectrum to the Schreierspectrum

Note that F takes points in the auxiliary spectrum to points inthe auxiliary spectrum. Our new goal is to map the auxiliaryspectrum to the Schreier spectrum.

R2 R2

R R

F

Ψ Ψ

f

Formally, we want to find Ψ and f that make this diagram commute.

As it turns out, Ψ and f do exist, and fpxq “ x2´x´3. [GS08]If x P σp∆npxqq, then fpxq P σp∆n´1pxqq. We can use this factto find the spectrum for any level.

18

Future research

Extend a theorem of Malozemov and Teplyaev [MT03] aboutgluing spectrally similar objects to self similar groups.

Extend a theorem of Nekrashevych and Teplyaev [NT08] aboutsymmetries of an object and spectral similarity to self similargroups.

Unify the above results to explain when a semi-conjugacy mapfor the Grigorchuk method exists and when it does not.

19

Thank you!

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References I

Rostislav Grigorchuk, Volodymyr Nekrashevych, andZoran Sunic. “From self-similar groups to self-similar setsand spectra”. In: Fractal geometry and stochastics V.Vol. 70. Progr. Probab. Birkhauser/Springer, Cham, 2015,pp. 175–207. doi: 10.1007/978-3-319-18660-3_11. url:https://doi.org/10.1007/978-3-319-18660-3_11.

Rostislav Grigorchuk and Zoran Sunik. “Asymptoticaspects of Schreier graphs and Hanoi Towers groups”. In:C. R. Math. Acad. Sci. Paris 342.8 (2006), pp. 545–550.issn: 1631-073X. doi: 10.1016/j.crma.2006.02.001.url: https://doi.org/10.1016/j.crma.2006.02.001.

Rostislav Grigorchuk and Zoran Sunic. “Schreier spectrumof the Hanoi Towers group on three pegs”. In: Analysis ongraphs and its applications. Vol. 77. Proc. Sympos. PureMath. Amer. Math. Soc., Providence, RI, 2008,pp. 183–198.

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References II

Daniel J. Kelleher, Benjamin A. Steinhurst, andChuen-Ming M. Wong. “From self-similar structures toself-similar groups”. In: Internat. J. Algebra Comput. 22.7(2012), pp. 1250056, 16.

Leonid Malozemov and Alexander Teplyaev.“Self-similarity, operators and dynamics”. In: Math. Phys.Anal. Geom. 6.3 (2003), pp. 201–218. issn: 1385-0172. doi:10.1023/A:1024931603110. url:https://doi.org/10.1023/A:1024931603110.

Volodymyr Nekrashevych. Self-similar groups. Vol. 117.Mathematical Surveys and Monographs. AmericanMathematical Society, Providence, RI, 2005, pp. xii+231.

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References III

Volodymyr Nekrashevych and Alexander Teplyaev.“Groups and analysis on fractals”. In: Analysis on graphsand its applications. Vol. 77. Proc. Sympos. Pure Math.Amer. Math. Soc., Providence, RI, 2008, pp. 143–180. doi:10.1090/pspum/077/2459868. url:https://doi.org/10.1090/pspum/077/2459868.

Robert S. Strichartz. Differential equations on fractals. Atutorial. Princeton University Press, Princeton, NJ, 2006,pp. xvi+169.

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