TWISTED BURNSIDE THEOREM FOR TYPE II_{1} GROUPS: AN EXAMPLE

ALEXANDER FEL’SHTYN, EVGENIJ TROITSKY, AND ANATOLY VERSHIK

Abstract. The purpose of the present paper is to discuss the following conjecture, which is a generalization of the classical Burnside theorem.

Let G be a countable discrete group, ϕ its automorphism, R(ϕ) the number of ϕconjugacy classes, S(ϕ) = # Fix(ϕ̂) the number of ϕ-invariant equivalence classes of irreducible unitary representations. If one of R(ϕ) and S(ϕ) is finite, then it is equal to the other.

This conjecture plays a very important role in the theory of twisted conjugacy classes having a long history (see [7], [3]) and has very serious consequences in Dynamics, while its proof needs rather fine results from Functional and Non-commutative Harmonic Analysis. It was proved for finitely generated groups of type I in [5].

In the present paper this conjecture is disproved for non-type I groups. More precisely, an example of a group and its automorphism is constructed such that the number of fixed irreducible representations is greater than the Reidemeister number. But the number of fixed finite-dimensional representations (i.e. the number of invariant finite-dimensional characters) in this example coincides with the Reidemeister number.

The directions for search of an appropriate formulation are indicated (another definition of the dual object).

Let G be a semidirect product of ℤ^{2} and ℤ by Anosov automorphism α with the matrix

A = ( \begin{matrix}2 & 1 \\ 1 & 1 \end{matrix} ). It consists by the definition of triples ((m, k), n) of integers with the

following multiplication low:

((m, k), n) ∗ ((m′, k′), n′) = ((m, k) + α^{n}(m′, k′), n + n′).

In particular,

((m, k), 0) ∗ ((0, 0), n) = ((m, k), n).

The inverse of ((m, k), n) is (−α^{−n}(m, k), −n). Indeed,

((m, k), n) ∗ (−α^{−n}(m, k), −n) = ((m, k) − α^{n}α^{−n}(m, k), n − n) = ((0, 0), 0).

The group G is a solvable (hence, amenable) group which is not of type I. Its regular representation is factorial. The irreducible representations can be obtained from ergodic orbits of the action of α on the torus 𝕋^{2} which is dual for the normal subgroup ℤ^{2} using appropriate cocycles [2, Sec. II.4], [1, Ch. 17, § 1].

Let us define an automorphism ϕ : G → G by

ϕ((m, k), n) = ((k, −m), −n),

This a joint research completed during the stay at Max-Planck-Institut für Mathematik (Bonn) in Spring 2004 in partial relation to the activity on Algebraic and Topological Dynamics. The authors are grateful to MPIM for support and hospitality. The second author is partially supported by RFFI Grant 02-01-00572, Grant for the support of leading scientific schools HIII-619.203.1 and Grant “Universities of Russia.”

i.e. the action on ℤ^{2} is defined by automorphism μ with the matrix M = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ),

and on ℤ by n ↦ −n. The map ϕ is clearly a bijection,

ϕ(((m, k), n) ∗ ((m′, k′), n′) = ϕ((m, k) + α^{n}(m′, k′), n + n′)) = ϕ((k, −m) + μα^{n}(m′, k′), −n − n′),

ϕ((m, k), n) ∗ ϕ(((m′, k′), n′) = ((k, −m), −n) ∗ ((k′, −m′), −n′)

= ((k, −m) + α^{−n}(k′, −m′), −n − n′).

Hence, to prove that ϕ is an automorphism, we need μα^{n} = α^{−n}μ. This follows from μα = α^{−1}μ. The further results in this direction can be found in [6].

Let us find the Reidemeister classes of ϕ, i.e. the classes of the equivalence relation h ∼ ghϕ(g^{−1}). For h = ((m, k), n) and g = ((x, y), z) the right hand side of the relation takes the following form:

((x, y) + α^{z}(m, k), z + n) ∗ (−μα^{−z}(x, y), z) =
((x, y) + α^{z}(m, k) − α^{z+n}μα^{−z}(x, y), 2z + n) =
(1)  = (α^{z}{(m, k) + (Id −α^{n}μ)α^{−z}(x, y)}, 2z + n).

Let us call level n (of G) the coset L_{n} of ℤ^{2} ⊂ G of all elements of the form ((m, k), n). Let us first take an element ((m, k), 0) from the level 0 and describe elements from the same level, being equivalent to it. By (1) in this case z = n = 0 and they have the form

((m, k) + (Id −μ)(x, y), 0) = ((m + (x − y), k + (x + y)), 0),

where Id −μ has the matrix ( \begin{matrix}1 & −1 \\ 1 & 1 \end{matrix} ). Hence, the level 0 has intersections with 2

Reidemeister classes, say, B_{1} and B_{2}. The first intersection B_{1} ∩ L_{0} is formed by elements ((u, v), 0) with odd u + v, and B_{2} ∩ L_{0} — with even u + v. The elements from the other levels, which are equivalent to ((m, k), 0), have the form

(2)  (α^{z}{(m, k) + (Id −μ)α^{−z}(x, y)}, 2z).

This means that B_{1} and B_{2} enter only even levels. Also, since α is an automorphism, we can rewrite (2) as

(3)  (α^{z}{(m, k) + (Id −μ)(u, v)}, 2z).

with arbitrary integers u and v. This means, that the intersections B_{i} ∩L_{2z} have the form α^{z}(B_{i}), i = 1, 2. In particular, the other Reidemeister classes do not enter even levels.

In a similar way, the elements of L_{1} equivalent to ((m, k), 1) have the form

((m, k) + (Id −αμ)(x, y), 1) = ((m + (2x − 2y), k + x), 1).

This means, that L_{1} enters 2 classes: B_{3} is formed by elements with even first coordinate, and B_{4} — with odd one. The elements from the other levels, which are equivalent to ((m, k), 1), have the form

(4)  (α^{z}{(m, k) + (Id −αμ)α^{−z}(x, y)}, 2z + 1).

Since α is an automorphism, we can rewrite (4) as

(5)  (α^{z}{(m, k) + (Id −αμ)(u, v)}, 2z + 1).

with arbitrary integers u and v. This means, that the intersections B_{i} ∩ L_{2z+1} have the form α^{z}(B_{i}), i = 3, 4. In particular, these four classes cover G and R(ϕ) = 4.

To obtain a complete description of B_{i} let us remark that directly from the definition of α

α(x, y) = (2x + y, x + y)

one has the following properties:

• α maps the set of elements with an even (resp., odd) sum of coordinates onto the set of elements with an even (resp., odd) second coordinate,

• α maps the set of elements with an even (resp., odd) second coordinate onto the set of elements with an even (resp., odd) first coordinate,

• α maps the set of elements with an even (resp., odd) first coordinate onto the set

of elements with an even (resp., odd) sum of coordinates. Hence, the elements ((m, k), n) in intersections B_{i} ∩ L_{j} are of the form

i  1  2  3  4
j ≡ 0 mod 6 m + k is even m + k is odd  ∅  ∅
j ≡ 1 mod 6  ∅  ∅  m is even  m is odd
j ≡ 2 mod 6 k is even  k is odd  ∅  ∅
j ≡ 3 mod 6  ∅  ∅  m + k is even m + k is odd
j ≡ 4 mod 6 m is even  m is odd  ∅  ∅
j ≡ 5 mod 6  ∅  ∅  k is even  k is odd

Now we want to study the fixed points of the homeomorphism ϕ̂: Ĝ → Ĝ, [ρ] ↦ [ρϕ] of the unitary dual. Let us start from the finite-dimensional representations. As it was shown in [5] in this case there exists exactly one twisted-invariant functional on L^{1}(G), or ϕ-central L^{∞} function, coming from a twisted-invariant functional on ρ(L^{1}(G)) ≅ M(dim ρ, ℂ) (up to scaling), namely

(6)  φ_{ρ} : g ↦ Tr(Sρ(g)),

where S is the intertwining operator between ρ and ρϕ.

First, we have to find μ-invariant finite α-orbits on 𝕋^{2}. One can notice that

det(A^{n} − M) = det A^{n} + 1 = 2

for any n. Hence, the mentioned orbits are formed by points with coordinates 0 and 1/2. We have 2 orbits: one of them consists of 1 point (0, 0) and gives rise to 1-dimensional trivial representation ρ_{1}, and the other consists of A_{1} = (0, 1/2), A_{2} = (1/2, 0) and A_{3} = (1/2, 1/2) and gives rise to a 3-dimensional (irreducible) representation ρ_{2}. Also, one has the following 1-dimensional representation π:

π((m, k), 2n) = 1, π((m, k), 2n + 1) = −1.

So, we have 4 representations

ρ_{1}, ρ_{2}, π, ρ_{2} ⊗ π.

We claim that they give rise via (6) to 4 linear independent twisted-invariant functionals. In particular, there is no more finite-dimensional fixed points of ϕ̂.

Clearly,

(7)  φ_{ρ_{1}} ≡ 1, φ_{π} = \{ \begin{matrix}1, & on ∪_{n} L_{2n} = B_{1} ∪ B_{2}, \\ −1, & on ∪_{n} L_{2n+1} = B_{3} ∪ B_{4}, \end{matrix} φ_{ρ_{2}⊗π} = φ_{ρ_{2}} · φ_{π}.

Let us find φ_{ρ_{2}}. In the space L^{2}(A_{1}, A_{2}, A_{3}) we take the base ε_{1}, ε_{2}, ε_{3} of characteristic functions in these points. One has

α(ε_{1}) = ε_{3},  α(ε_{3}) = ε_{2},  α(ε_{2}) = ε_{1},
μ(ε_{1}) = ε_{2},  μ(ε_{2}) = ε_{1},  μ(ε_{3}) = ε_{3}.

The representation (see [1, Ch. 17, § 1]) is defined by:

ρ_{2}(m, k, 0)(ε_{i}) = χ_{A_{i}}(m, k) · ε_{i}, ρ_{2}(0, 0, n)(ε_{i}) = α^{−n}(ε_{i}) = ε_{i+n mod 3},
ρ_{2}(m, k, 0)(ε_{1}) = e^{2πi(0·m+1/2·k)} · ε_{1} = e^{πik}ε_{1},
ρ_{2}(m, k, 0)(ε_{2}) = e^{2πi(1/2·m+0·k)} · ε_{2} = e^{πim}ε_{2},
ρ_{2}(m, k, 0)(ε_{3}) = e^{2πi(1/2·m+1/2·k)} · ε_{3} = e^{πi(m+k)}ε_{3}.

The representation ϕ̂ρ_{2} is defined by

ϕ̂ρ_{2}(0, 0, n)(ε_{i}) = ρ_{2}(0, 0, −n)(ε_{i}) = ε_{i−n mod 3},
ϕ̂ρ_{2}(m, k, 0)(ε_{1}) = ρ_{2}(k, −m, 0)(ε_{1}) = e^{−πim}ε_{1} = e^{πim}ε_{1},
ϕ̂ρ_{2}(m, k, 0)(ε_{2}) = ρ_{2}(k, −m, 0)(ε_{2}) = e^{πik}ε_{2},
ϕ̂ρ_{2}(m, k, 0)(ε_{3}) = ρ_{2}(k, −m, 0)(ε_{3}) = e^{πi(k−m)}ε_{3} = e^{πi(m+k)}ε_{3}.

The intertwining operator is induced by ϕ and has the matrix S = ( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ). Hence,

φ_{ρ_{2}}(m, k, n) = Tr(Sρ_{2}(m, k, 0)ρ_{2}(0, 0, n)) =

= Tr [( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ) ( \begin{matrix}e^{πik} & 0 & 0 \\ 0 & e^{πim} & 0 \\ 0 & 0 & e^{πi(k+m)} \end{matrix} ) ( \begin{matrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} )^{n}] =

= Tr [( \begin{matrix}0 & e^{πik} & 0 \\ e^{πik} & 0 & 0 \\ 0 & 0 & e^{πi(k+m)} \end{matrix} ) ( \begin{matrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} )^{n}] .

For n ≡ 0 mod 3

φ_{ρ_{2}}(m, k, n) = Tr [( \begin{matrix}0 & e^{πim} & 0 \\ e^{πik} & 0 & 0 \\ 0 & 0 & e^{πi(k+m)} \end{matrix} )] = e^{πi(k+m)} = \{ \begin{matrix}1, & if m + k is even \\ −1, & if m + k is odd \end{matrix} ,

for n ≡ 1 mod 3

φ_{ρ_{2}}(m, k, n) = Tr [( \begin{matrix}0 & e^{πim} & 0 \\ e^{πik} & 0 & 0 \\ 0 & 0 & e^{πi(k+m)} \end{matrix} ) ( \begin{matrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} )] =

= Tr ( \begin{matrix}e^{πim} & 0 & 0 \\ 0 & 0 & e^{πik} \\ 0 & e^{πi(k+m)} & 0 \end{matrix} ) = e^{πim} = \{ \begin{matrix}1, & if m is even \\ −1, & if m is odd \end{matrix} ,

for n ≡ 2 mod 3

φ_{ρ_{2}}(m, k, n) = Tr [( \begin{matrix}0 & e^{πim} & 0 \\ e^{πik} & 0 & 0 \\ 0 & 0 & e^{πi(k+m)} \end{matrix} ) ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix} )] =

= Tr ( \begin{matrix}0 & 0 & e^{πim} \\ 0 & e^{πik} & 0 \\ e^{πi(k+m)} & 0 & 0 \end{matrix} ) = e^{πik} = \{ \begin{matrix}1, & if k is even \\ −1, & if k is odd \end{matrix} .

φ_{ρ_{2}} is 3-periodical in n, while the characteristic functions of B_{i} are 6-periodical. For j = 0, …, 5 one has

\begin{matrix}φ_{ρ_{2}}|_{B_{1}∩L_{0}} ≡ 1, & φ_{ρ_{2}}|_{B_{2}∩L_{0}} ≡ −1, & φ_{ρ_{2}}|_{B_{3}∩L_{1}} ≡ 1, & φ_{ρ_{2}}|_{B_{4}∩L_{1}} ≡ −1, \\ φ_{ρ_{2}}|_{B_{1}∩L_{2}} ≡ 1, & φ_{ρ_{2}}|_{B_{2}∩L_{2}} ≡ −1, & φ_{ρ_{2}}|_{B_{3}∩L_{3}} ≡ 1, & φ_{ρ_{2}}|_{B_{4}∩L_{3}} ≡ −1, \\ φ_{ρ_{2}}|_{B_{1}∩L_{4}} ≡ 1, & φ_{ρ_{2}}|_{B_{2}∩L_{4}} ≡ −1, & φ_{ρ_{2}}|_{B_{3}∩L_{5}} ≡ 1, & φ_{ρ_{2}}|_{B_{4}∩L_{5}} ≡ −1, \end{matrix}

so φ_{ρ_{2}}|_{B_{1}∪B_{3}} ≡ 1, φ_{ρ_{2}}|_{B_{2}∪B_{4}} ≡ −1. the determinant of the values of the functions φ_{ρ_{1}}, φ_{π}, φ_{ρ_{2}}, φ_{ρ_{2}⊗π} on B_{1}, B_{2}, B_{3}, B_{4} is

det ( \begin{matrix}1 & 1 & 1 & 1 \\ 1 & 1 & −1 & −1 \\ 1 & −1 & 1 & −1 \\ 1 & −1 & −1 & 1 \end{matrix} ) = −8 ≠ 0.

Hence, they are linearly independent.

Nevertheless, there are infinite-dimensional irreducible ϕ̂-invariant representations. E.g. we have a representation ρ of G on L^{2}(𝕋^{2}) with the respect to the Lebesgue measure, with ρ(m, k, 0) be the multiplier by characters in the appropriate points and ρ(0, 0, 1) is α, in the same manner as for ρ_{2}.

This disproves the conjecture of Fel’shtyn and Hill [4], who supposed that the Reidemeister number equals to the number of fixed points of ϕ̂ on Ĝ

This representation is not traceable, but one can nevertheless try to calculate (6). Let us choose an orthonormal base of L^{2}(𝕋^{2}) formed by ε_{st}(x, y) = e^{2πi(sx+ty)}, x, y ∈ [0, 1]. The intertwining operator S is generated by μ. Then

Tr(Sρ(m, k, n)) = ∑_{s,t} ∫_{0}^{1} ∫_{0}^{1}(ρ(m, k, 0)ρ(0, 0, n)ε_{s,t})(μ(x, y))\overline{ε_{s,t}}(x, y) dx dy =

= ∑_{s,t} ∫_{0}^{1} ∫_{0}^{1} e^{2πi⟨(m,k),μ(x,y)⟩}(ρ(0, 0, n)ε_{s,t})(μ(x, y))ε_{−s,−t}(x, y) dx dy =

= ∑_{s,t} ∫_{0}^{1} ∫_{0}^{1} e^{2πi⟨(m,k),μ(x,y)⟩}ε_{s,t}(α^{n}μ(x, y))ε_{−s,−t}(x, y) dx dy =

= ∑_{s,t} ∫_{0}^{1} ∫_{0}^{1} e^{2πi⟨(m,k)+(α^{n}−μ)(s,t),μ(x,y)⟩} dx dy = ∑_{(m,k)=(μ−α^{n})(s,t)} 1.

For n = 0 this equals 1 for m + k even and 0 for m + k odd. Hence, φ_{ρ}|_{B_{1}∩L_{0}} = 1, φ_{ρ}|_{B_{2}∩L_{0}} = 0. For n = 1 the equality takes the form (m, k) = (−2s − 2t, −t). Hence, φ_{ρ}|_{B_{3}∩L_{1}} = 1, φ_{ρ}|_{B_{4}∩L_{1}} = 0. If n = 2r, the equality takes form

(m, k) = (μ − α^{2r})(s, t), (m, k) = (α^{r}μα^{r} − α^{2r})(s, t),
α^{−r}(m, k) = (μ − α^{0})α^{r}(s, t).

Since α is an automorphism of ℤ ⊕ ℤ and by the description of B_{i} via the action of α, we obtain that φ_{ρ}|_{B_{1}} = 1, φ_{ρ}|_{B_{2}} = 0. Similarly, for n odd. So, φ_{ρ} is well defined and

φ_{ρ} = χ_{B_{1}} + χ_{B_{3}} = \frac{1}{2}(φ_{ρ_{1}} + φ_{ρ_{2}}).

Of course, there are also ϕ̂-invariant traceable factor representations of this group G, e.g. the regular representation. Since its kernel is trivial, evidently all twisted-invariant functionals can be pushed back from it.

This example shows that the conjecture of [4] for general groups can survive only after eliminating badly separated points in Ĝ.

References

[1] Asim O. Barut and Ryszard R‘aczka, Theory of group representations and applications, second ed., World Scientific Publishing Co., Singapore, 1986.

[2] Alain Connes, Noncommutative geometry, Academic Press Inc., San Diego, CA, 1994.

[3] Alexander Fel’shtyn, Dynamical zeta functions, Nielsen theory and Reidemeister torsion, Mem. Amer. Math. Soc. 147 (2000), no. 699, xii+146.

[4] Alexander Fel’shtyn and Richard Hill, The Reidemeister zeta function with applications to Nielsen theory and a connection with Reidemeister torsion, K-Theory 8 (1994), no. 4, 367–393.

[5] Alexander Fel’shtyn and Evgenij Troitsky, A twisted Burnside theorem for countable groups and Reidemeister numbers, Noncommutative Geometry and Number Theory (Bonn, 2003), Vieweg, Braunschweig, 2004, pp. ??–??

[6] Daciberg Gonçalves and Peter Wong, Twisted conjugacy classes in exponential growth groups, Bull. London Math. Soc. 35 (2003), no. 2, 261–268.

[7] B. Jiang, Lectures on Nielsen fixed point theory, Contemp. Math., vol. 14, Amer. Math. Soc., Providence, RI, 1983.

Fachbereich Mathematik, Emmy-Noether-Campus, Universität Siegen, Walter-Flex-Str. 3, D-57068 Siegen, Germany and Instytut Matematyki, Uniwersytet Szczecinski, ul. Wielkopolska 15, 70-451 Szczecin, Poland

  E-mail address: felshtyn@math.uni-siegen.de
  URL: http://www.math.uni-siegen.de/felshtyn

Dept. of Mech. and Math., Moscow State University, 119992 GSP-2 Moscow, Russia

  E-mail address: troitsky@mech.math.msu.su
  URL: http://mech.math.msu.su/~troitsky

St.Petersburg Department of Steklov Institute of Mathematics, 27 Fontanka, St. Petersburg 191011, Russia

  E-mail address: vershik@pdmi.ras.ru
  URL: http://www.pdmi.ras.ru/~vershik