Diestel-Leader Groups Jennifer Taback Bowdoin College Joint work - - PowerPoint PPT Presentation

Automorphisms of Diestel-Leader Groups

Jennifer Taback Bowdoin College

Joint work with Melanie Stein and Peter Wong GAGTA Conference May 28, 2013

Diestel-Leader Groups

We define the Diestel-Leader group Γ𝑒(𝑟) and generating set 𝑇𝑒(𝑟) so that the resulting Cayley graph is the Diestel-Leader graph 𝐸𝑀𝑒 𝑟 :

𝐸𝑀𝑒 𝑟 ⊆ 𝑈

1 × 𝑈2 × ⋯ × 𝑈𝑒−1 × 𝑈𝑒

t+𝑚1 t+𝑚2 t+𝑚𝑒−1 𝑢−1 Formal Variables: Where the 𝑚𝑗 ∈ 𝑎𝑟 are chosen so that 𝑚𝑗 − 𝑚𝑘 ∈ 𝑎𝑟

∗.

Generating set 𝑇𝑒 𝑟 : t+𝑚𝑗 𝑐 1

±1

and (t+𝑚𝑗)(t+𝑚𝑘)−1 −𝑐(t+𝑚𝑘 ) 1

±1

where b ∈ 𝑎𝑟 . This construction of the groups Γ𝑒(𝑟) is due to Bartholdi, Neuhauser and Woess and requires that if p is any prime divisor of q, then d ≤ p+1.

Diestel-Leader Groups

A generic element of Γ𝑒(𝑟) has the form g= (𝑢 + 𝑚𝑗)𝑓𝑗

𝑒−1 𝑗=1

𝑄 1 where 𝑓𝑗 ∈Z and P is a polynomial in 𝑎𝑟[(t + 𝑚1) −1, 𝑢 + 𝑚2 −1, ⋯,(t + 𝑚𝑒−1) −1, 𝑢]. Decomposition Lemma: Let g ∈ Γ𝑒(𝑟) be as above. Then P can be written uniquely as P=𝑄

1 + 𝑄2 + ⋯ + 𝑄𝑒−1 + 𝑄𝑒

𝑄𝑗 is a polynomial in t+𝑚𝑗 with

• nly negative degree terms.

𝑄𝑒 is a polynomial in 𝑢−1with only terms of non-positive degree.

Rough idea of identification between g and a vertex in 𝐸𝑀𝑒(𝑟): (𝑓1, 𝑓2, ⋯ , 𝑓𝑒−1) and P=𝑄

1 + 𝑄2 + ⋯ + 𝑄𝑒−1 + 𝑄𝑒 determine the vertex in

𝐸𝑀𝑒 𝑟 corresponding to g.

Algebraic comparison to Lamplighter groups

Lamplighter group:

𝑀𝑜 = 𝑎𝑜∫ Z = < a,t |𝑏𝑜=1, [𝑏𝑢𝑗, 𝑏𝑢𝑘]=1,i,j∈ 𝑎>

Elements of Ln are often represented by a finite number of illuminated “bulbs” along with an integral position of the “lamplighter.”

Configurations of illuminated bulbs Positions of the lamplighter

Algebraic comparison to Lamplighter groups

Diestel-Leader groups

Here L is the analog of the infinite string of “lamps” and is the union of d-1 rays in 𝑎𝑒−1.

Commutator subgroup: Configurations of “bulbs,” Vertices have all coordinates at height 0 in all d-1 trees Abelianization: Positions of “lamplighter,” Coordinates represent the heights In the first d-1 trees; height in the last tree is determined.

Automorphisms of Diestel-Leader groups

Induced automorphisms on the quotient

Proof that induced automorphisms of the quotient are permutation matrices

Proof that induced automorphisms of the quotient are permutation matrices

Upper right entry is l_j-l_i

Question: What is 𝑄𝑒?

Upper right entry Is a big mess! Denote it by S-T.

Proof that induced automorphisms of the quotient are permutation matrices

Understanding the commutator subgroup Γ𝑒(𝑟)′

The commutator subgroup of Γ𝑒(𝑟)

In general, the commutator subgroup of Γ𝑒 𝑟 is defined by

View as indexing conjugates of 1 1 1 by products of 𝑢 + 𝑚𝑗 1

𝑓𝑗

where K is generated by simply polynomial relations between the formal Variables of the form:

Further restriction of induced automorphisms

• f 𝑎𝑒−1

From the split short exact sequence we see that if Suppose that φ corresponds to a permutation in Σ𝑒which contains the cycle (1 2 3 ⋯ 𝑙). Then φ transforms the relations 𝐿1,2, 𝐿2,3, ⋯ , 𝐿𝑙,1 into a system of equations which has a solution iff k=d-1=q is prime. ∎ then

Defining relations of the kernel

Counting twisted conjugacy classes in Γ𝑒(q)

If ϕ=Id then this is regular conjugacy.

Counting twisted conjugacy classes in Γ𝑒(q)

If ϕ=Id then this is regular conjugacy.

2 is SMALL! d≥ 𝟒 𝒋𝒕 𝑴𝑩𝑺𝑯𝑭 𝑺 !

Counting twisted conjugacy classes in Γ𝑒(q)

The one diagram proof… When φ is a permutation matrix it is easy to show that R(φ )=∞ and hence that R(ϕ)=∞. When d=3 and q=2 we can verify this directly for the 4 remaining matrices.