II1 -factor representations of Higman-Thompson’s
groups
Classification of Characters
Kostya Medynets
United States Naval Academy
Annapolis, Maryland
Characters of infinite groups
Fix an infinite group G .
Definition
A character of G is a function f : G → C such that
(normalization) f (1) = 1
(class function) f (a) = f (bab −1 ) for all a, b ∈ G .
(positivity) For any family {g1 , . . . , gn } ⊂ G , the matrix
M = {f (gi−1 gj )}ni,j=1 is positive-definite, i.e., for any
{c1 , . . . , cn } ⊂ C
n
n X
X
i =1 j=1
c̄i cj f (gi−1 gj ) ≥ 0.
Where do the characters come from?
Consider a unitary representation π : G → B(H) in a Hilbert space
H. Denote by Wπ the von Neumann algebra generated by π(G ),
i.e., Wπ = (π(G ))′′ . Suppose that Wπ is of finite type, of type In ,
n < ∞, or II1 .
Let τ be the tracial state on Wπ . Set f (g ) = τ (π(g )). Then f is a
character.
To show
P notice that τ (Q) = (Qξ, ξ).
P that f is positive-definite,
Then ni,j=1 c̄i cj f (gi−1 gj ) = || ci π(gi )ξ||2 for any c1 , . . . , cn .
Do we have any other characters?
Theorem (Gelfand-Naimark-Segal Construction)
If f is a character of an infinite group G , then there is a finite type
unitary representation π such that f (g ) = τ (π(g )), where τ is the
trace of von Neumann algebra Wπ .
Characters form a Choquet simplex.
(Decomposition) Extreme characters correspond to factor
representations. This means that the algebra generated by the
group π(G ) is a factor.
So the question is to describe all extreme characters for a group in
question.
For finite groups, extreme characters are precisely the irreducible
characters.
Characters and Ergodic Theory
Vershik noticed that there is yet another way to produce
characters. Suppose that a group G acts on a measure space
(X , µ). For g ∈ G , set Fix(g ) = {x ∈ X : g (x) = x}, the set of
fixed points. The function f (g ) = µ(Fix(g )) is a character.
The function f is central.
f (hgh−1 ) = µ(Fix(hgh−1 )) = µ(h(Fix(g ))) = µ(Fix(g )) = f (g ).
Normalization. f (1) = µ(X ) = 1.
The function f is positive-definite.
Characters and Ergodic Theory
To show that f is positive-definite, we will use the Feldman-Moore
groupoid construction.
Let R be the orbit equivalence relation generated by the action of
G on (X , µ).
R
Define the measure µ̄ on R as µ(A) = X card(Ax )dµ(x), where
Ax = {(x, y ) ∈ A}.
Define the unitary representation π : G → U(L2 (R, µ̄)) by
π(g )f (x, y ) = f (g −1 x, y ).
Denote by ∆ the diagonal in X × X . Then
f (g ) = µ(Fix(g )) = (π(g )1∆ , 1∆ ).
Thompson’s group
Thompson’s group F is generated by the functions
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
The group F consists of all piecewise linear homeomorphisms from
the closed interval [0, 1] to itself that are differentiable except at
finitely many dyadic rational numbers and such that on the
intervals of differentiability the derivatives are powers of 2. The
elements of F preserve orientation.
The Commutator Subgroup
Denote by F ′ the commutator subgroup of F . The group F ′
consists of all homeomorphisms from F that are trivial in
neighborhoods of 0 and 1.
F ′ is simple.
The action of F ′ on X = (0, 1) has no probability invariant
measure.
Theorem (Dudko-M.)
The group F ′ has no non-trivial characters.
Every group always admits at least two characters (1) the regular
character δ(g ) = 0 if g 6= e and δ(e) = 0 and (2) the identity
character ρ(g ) = 1 for every g .
Corollaries
Theorem (Dudko-M.)
The only non-trivial II1 -factor representation of F ′ is the left
regular representation.
Theorem (Dudko-M.)
If a countable group G has no non-trivial characters, then any
ergodic action of G on a probability measure space is essentially
free, i.e. µ(Fix(g )) = 0 for every g 6= e.
Invariant Random Subgroups
Definition
Every group G acts on the space SUBG of its subgroups by
conjugation. The space of subgroups is a closed subset in {0, 1}G .
A probability invariant measure µ for (SUBG , G ) is called an
invariant random subgroup.
Abert, Glasner, and Viraǵ (2012) showed that for any invariant
random subgroup λ there is a measure preserving action of G on
(X , µ) such that F : X → SUBG given by
F (x) = StabG (x) = {g ∈ G : g (x) = x} pushes µ onto λ.
Corollary
The commutator subgroup F ′ of the Thompson group has no
non-trivial invariant random subgroups. Note that G always has
measures supported by 1{e} and 1G .
Amenability and Invariant Random Subgroups
Open Question. Is F ′ amenable?
A discrete group G is amenable if it admits an invariant finitely
additive probability measure on G .
Question. Is there a subgroup Γ < F ′ such that the orbit-closure of
Γ, under F ′ , in SUBG does not contain the identity subgroup {e}
and the entire group {F ′ }? If it were true, the group F ′ would be
non-amenable. Indeed, suppose that F ′ is amenable. Then there
exists an invariant random subgroup µ supported by the
orbit-closure of Γ. This is impossible as F ′ has no non-trivial
invariant random subgroups.
Skudlarek (1971). There are locally finite groups with no
non-trivial characters.
Characters of Thompson’s groups
The abelianization of F , F /F ′ is isomorphic to Z × Z.
Theorem (Dudko-M.)
If χ : F → C is an irreducible character, then either χ is a regular
character or χ(g ) = ρ([g ]), where ρ : F /F ′ → T is a group
homomorphism, where [g ] is the image of g in the abelianization
F /F ′ .
Proof. Classification of Characters for F ′ .
Simplicity. Note that the group F ′ is simple.
If χ is a character such that |χ(g )| = 1 for some g ∈ G ,
g 6= e, then χ ≡ 1.
Using the GNS-construction, construct a unitary
representation (π, H, ξ) such that χ(g ) = (π(g )ξ, ξ) with
||ξ|| = 1. Using the Cauchy-Schwarz inequality, we obtain
1 = |χ(g )| = |(π(g )ξ, ξ)| ≤ 1.
Thus, π(g )ξ = cξ. Since ξ is a separating, we get that
π(g ) = cId. Find h ∈ F ′ such that s = hgh−1 g −1 6= e. Then
π(s) = Id.
Since the group F ′ is simple, it is generated by elements of
the form {qsq −1 }. It follows that π ≡ 1.
Proof continued.
If χ is irreducible, the representation is a factor, then χ is
multiplicative in the sense that for “almost” any g , h ∈ F ′
with disjoint supports, χ(g · h) = χ(g ) · χ(h).
The action of the group F ′ on X = (0, 1) is contractible in
the sense that for any g ∈ F ′ and any open set U there is a
conjugate of g supported by U.
So for any open set U and any ε > 0, there is an element
h ∈ F ′ supported by U with |χ(h)| < ε. Take |χ(g )| < 1, find
conjugates {gi } of g with disjoint supports and supported by
U. Then |χ(g1 · · · gn )| < ε.
If χ(g ) = µ(Fix(g )). Then supp(gn ) → {x0 }, Fix(gn ) → X ,
µ(Fix(gn )) → 1, but χ(gn ) → 0, a contradiction.