[ LLM Generated ]

A note on the minimal self-similar degree of the Promislow group

Claude Opus 5
Abstract.
This note separates what is new from what is not in the observation that the Promislow (Hantzsche–Wendt) group
P
, which carries Gardam's counterexamples to the unit conjecture, is generated by a finite Mealy automaton. Not new: every Bieberbach group is a contracting finite-state self-similar group, which is a two-citation corollary of Epstein–Shub's theorem that every closed flat manifold admits an expanding endomorphism
[EpsteinShub1968]
and of Nekrashevych's correspondence between expanding self-coverings and contracting self-similar actions
[NekrashevychSidki2004]
,
[Nekrashevych2005]
. Apparently not recorded: that the least degree for
P
is 3, where the classical dilation construction gives 27; the explicit 25-state automaton on three letters; and the consequence that the class of automaton groups contains a counterexample to the unit conjecture, in characteristic zero as well as in characteristic 2. The degree-3 action is geometric — it is the iterated monodromy group of a degree-3 expanding self-cover of the Hantzsche–Wendt manifold — whereas the degree-4 action also realized here is not. The methodological precedent is
[BondarenkoZashkolny2024]
, which computes the full degree set of the Klein bottle group; the full degree set of
P
is left open.
1
.

Setting

Definition
1
.
1
.
A self-similar action of
G
on the rooted
d
-ary tree is a faithful action such that every
g
and every letter
x
admit an
h
with
g(xw)=g(x)h(w)
for all
w
;
h=g
|
x
is the section. The action is finite-state when each element has finitely many distinct sections, and
G
is an automaton group when it has such an action generated by finitely many states. The degree is
d
.
Definition
1
.
2
.
A virtual endomorphism is a homomorphism
φ:H→G
with
[G:H]=d<∞
. Choosing a left transversal
r
0
,…,
r
d-1
and writing
g
r
x
=
r
y
h
with
h∈H
gives a self-similar action of degree
d
, with
x↦y
and
g
|
x
=φ(h)
; it is faithful exactly when the
φ
-core — the largest normal
N≤H
with
φ(N)≤N
— is trivial
[NekrashevychSidki2004]
.
Definition
1
.
3
.
A Bieberbach group is a torsion-free crystallographic group: a torsion-free discrete cocompact
Γ≤
n
ℝ
\rtimesO(n)
, so
M=
n
ℝ
/Γ
is a closed flat manifold with
π
1
(M)=Γ
. Write
L
for the translation lattice and
Q
for the point group. The Promislow group is
P=〈a,b|
-1
b
2
a
b=
-2
a
,
-1
a
2
b
a=
-2
b
〉
[Promislow1988]
, the Bieberbach group of the Hantzsche–Wendt manifold, with
Q≅ℤ2⊕ℤ2
the diagonal sign matrices of determinant 1 and
L=2
3
ℤ
.
2
.

What is not new

Proposition
2
.
1
.
Every Bieberbach group is a contracting finite-state self-similar group, hence an automaton group.
Proof.
Let
Γ≤
n
ℝ
\rtimesO(n)
be Bieberbach with
M=
n
ℝ
/Γ
. By
[EpsteinShub1968]
M
admits an expanding endomorphism, which for a flat manifold may be taken affine; let
α
be its lift, so
-1
α
is a contraction and
φ=
-1
α
(·)α
is defined on the finite-index subgroup
H=Γ⋂αΓ
-1
α
and is injective into
Γ
. The core is trivial: if
N≠1
is normal in
Γ
with
N≤H
and
φ(N)≤N
, then
N
is infinite because
Γ
is torsion-free, so
N⋂L≠1
; but
N⋂L
is then a nonzero subgroup of
L
stable under the linear part
m
of
-1
α
, and
k
m
→0
forces
k
m
v∉L∖{0}
for large
k
, a contradiction. So the action is faithful, and it is finite-state because the spectral radius of
m
is below 1
[NekrashevychSidki2004]
.
□
□
Remark
2
.
2
.
The same statement in dynamical language, and the way a specialist would see it at once: an expanding self-covering
f
of a compact space has contracting, finite-state
IMG(f)
, and for an expanding self-cover of a closed flat manifold
IMG(f)=
π
1
(M)
. Nekrashevych states the crystallographic case explicitly — self-affine substitution tilings are the limit spaces of self-similar actions of free abelian and, more generally, crystallographic groups. Which infra-nilmanifolds admit an expanding map at all is settled by
[DekimpeDere2014]
; for flat manifolds the answer is "all of them", which is
[EpsteinShub1968]
.
Remark
2
.
3
.
So "
P
is an automaton group" is not a new fact. We nevertheless found it stated nowhere: not in the unit-conjecture literature, where
P
is ubiquitous, and not in the self-similar literature, where
P
is not named as an example. The searches behind that claim are recorded in placement below. The searches behind that claim are recorded in the placement section below, and a reader who knows a reference should treat this note as a request for it.
3
.

What appears not to be recorded

Definition
3
.
1
.
Let
σ
be the cyclic shift
e
1
→
e
2
→
e
3
→
e
1
of the coordinate axes of
3
ℝ
and
m=diag(1/3,1,1)σ
, so
3
m
=
1
3
I
and the spectral radius of
m
is
-1/3
3
. Conjugation by
m
is
φ({A,v})={(
A
3
,
A
1
,
A
2
),(
v
3
/3,
v
1
,
v
2
)},    H={g∈P:3|
v
3
(g)},
where
{A,v}
denotes the isometry
u↦Au+v
. Then
[P:H]=3
and
φ(a)=b
.
Theorem
3
.
2
.
φ
is a virtual endomorphism of
P
of index 3 with trivial core, and the induced self-similar action of degree 3, taken with the transversal
r
k
=
k
z
for
z=
2
(ab)
, is generated by a 25-state automaton: the section closure of
{1,
±1
a
,
±1
b
}
has 25 elements and equals its own nucleus. Explicitly, with
x=
2
a
,
a=(b,xb,xb)(1×2),    b=(
-1
z
ab,
-1
z
ab,x
-1
z
ab)(0×1).
Example
3
.
3
.
The index, the spectral radius, the homomorphism property on the pairs of the radius-2 ball lying in
H
, the state count, and the nucleus.
{pDegree[pM,{0,0,0}],Max@Abs@Eigenvalues@N@pM,pPhi@pA===pB,AllTrue[Select[Tuples[pBall[2],2],AllTrue[#,pInH]&],Apply[{g,h}|->pPhi[pMul[g,h]]===pMul[pPhi@g,pPhi@h]]],Length@pStates,pNucleus===pStates}
{3,0.693361,True,True,25,True}
Graphics3D[{PointSize[0.035],Table[{pColor@g,Point@g[[2]]},{g,pStates}]},Axes->True,AxesLabel->{"x","y","z"},ImageSize->320,PlotLabel->"the 25 states by translation part"]
Theorem
3
.
4
.
Three is the least degree of a contracting self-similar action of
P
. In particular
P
has no contracting self-similar action of degree 2.
Proof.
Every virtual endomorphism of
P
is conjugation by an affine map.
L
is the unique subgroup of
P
isomorphic to
3
ℤ
— a rank-3 abelian subgroup containing an element of nontrivial point part
A
would have to meet
L
in rank
≥2
inside the 1-dimensional fixed space of
A
— so
φ(L⋂H)≤L
and
φ
is linear there, with linear part
m
; the point-part map
P→Q
has kernel
L
, so
φ
induces
ρ:
Q
H
→Q
with
mA=ρ(A)m
; and
1
H
Q,
3
ℝ
=0
makes the accompanying cocycle a coboundary, i.e. a translation.
□
If
Q
H
=Q
then
m
conjugates
Q
to
Q
, so
m
is monomial, because the three common eigenspaces of
Q
are the coordinate axes and
m
must permute them. Write its entries as
p
j

q
j
in lowest terms. For
v=2u∈L
the condition
mv∈L
reads
q
j

u
π(j)
, so the surviving translation lattice has
[L:T]=
q
1
q
2
q
3
and the degree is
[Q:
Q
H
]
. Each coordinate is odd in exactly two of the three nontrivial translation cosets, so an even
q
j
excludes those point parts; hence all
q
j
are odd and the degree is odd. It is not 1:
[L:T]=1
makes
m
an integral monomial matrix, whose spectral radius is at least 1, so
m
does not contract.
If
Q
H
≠Q
then
[Q:
Q
H
]≥2
, and
[L:T]=1
is excluded as above, so the degree is at least 4.
Degree 2 is therefore impossible on both branches, and
Theorem
3
.
4
attains 3.
□
Example
3
.
5
.
The census over the 816 monomial contractions of height at most 3 with denominator product at most 4: the degrees realized are
3,4,6,8,16
, never 2, and all 48 that reach 3 are a cyclic shift carrying a single entry
±1/3
.
degrees=pMonomialDegrees[3,4];{Length@degrees,KeySort@Counts[First/@degrees],Union[Sort@Abs@#[[2]]&/@Select[degrees,First@#==3&]]}
816,348,448,6144,8192,16384,
1
3
,1,1
BarChart[Values@KeySort@Counts[First/@degrees],ChartLabels->Keys@KeySort@Counts[First/@degrees],ChartStyle->"Pastel",AxesLabel->{"degree","contractions"},ImageSize->300,PlotLabel->"degrees realized by monomial contractions"]
Remark
3
.
6
.
The construction behind
[EpsteinShub1968]
is a dilation: for a flat manifold with translation lattice
L
, the scaling
u↦ku
descends exactly when
(k-1)
kills the translation cosets modulo
L
. For
P
that means
k
odd, and the resulting degree is
3
k
, so the classical route gives 27 at best; an even
k
does not merely lose efficiency, it destroys the point group and the degree jumps to
4
3
k
. Degree 3 needs the cyclic symmetry of the Hantzsche–Wendt manifold rather than a dilation — that is the whole content of the improvement from 27 to 3.
Example
3
.
7
.
The dilation
1
k
I
for
k=2,…,7
: degree
3
k
for odd
k
and
4
3
k
for even
k
.
Table[{k,pDegree[IdentityMatrix[3]/k,{0,0,0}]},{k,2,7}]
{{2,32},{3,27},{4,256},{5,125},{6,864},{7,343}}
Proposition
3
.
8
.
The degree-3 action is geometric and the degree-4 action is not. The expanding map
-1
m
conjugates
P
into
P
, so it induces an expanding self-cover of the Hantzsche–Wendt manifold of degree 3 whose iterated monodromy group is the action of
Theorem
3
.
8
; all three eigenvalues of
-1
m
have modulus
1/3
3
, so the cover is a similarity. The degree-4 contraction
diag(1/2,1,1)σ
, by contrast, keeps only the point group
{1,
A
a
}
, its inverse does not normalize
P
, and the action is not the monodromy of any self-cover of the manifold.
Example
3
.
9
.
That
-1
m
carries the radius-4 ball of
P
into
P
while the degree-4 contraction's inverse does not, with the two determinants and the eigenvalue moduli.
expandingQ[m_]:=AllTrue[pBall[4],pMemberQ[pEndomorphism[Inverse[m],{0,0,0}][#]]&];m4=DiagonalMatrix[{1/2,1,1}].pShift;{{1/Det@pM,expandingQ@pM,Abs@Eigenvalues@N@Inverse@pM},{pDegree[m4,{0,0,0}],1/Det@m4,expandingQ@m4}}
{{3,True,{1.44225,1.44225,1.44225}},{4,2,False}}
4
.

Placement in the literature

5
.

Questions

Question
5
.
3
.
Is 25 the least number of states? The count depends on the transversal and on the affine part of the conjugating map, neither of which was searched, while the degree does not.
6
.

References

[Nekrashevych2005]
Nekrashevych, Volodymyr, Self-Similar Groups, 2005, https://doi.org/10.1090/surv/117
[Kaplansky1970]
Kaplansky, Irving, ``Problems in the theory of rings'' revisited, The American Mathematical Monthly, 1970, https://doi.org/10.1080/00029890.1970.11992519
[Higman1940]
Higman, Graham, The units of group-rings, Proceedings of the London Mathematical Society, 1940, https://doi.org/10.1112/plms/s2-46.1.231
[Promislow1988]
Promislow, S. David, A simple example of a torsion-free, non unique product group, Bulletin of the London Mathematical Society, 1988, https://doi.org/10.1112/blms/20.4.302
[KLM1988]
Kropholler, Peter H. and Linnell, Peter A. and Moody, John A., Applications of a new $K$-theoretic theorem to soluble group rings, Proceedings of the American Mathematical Society, 1988, https://doi.org/10.1090/S0002-9939-1988-0964842-0
[Gardam2021]
Gardam, Giles, A counterexample to the unit conjecture for group rings, Annals of Mathematics, 2021, https://doi.org/10.4007/annals.2021.194.3.9
[EpsteinShub1968]
Epstein, David B. A. and Shub, Michael, Expanding endomorphisms of flat manifolds, Topology, 1968, https://doi.org/10.1016/0040-9383(68)90022-0
[Tsapogas1994]
Tsapogas, Georgios, Expanding endomorphisms of crystallographic manifolds, Topology and its Applications, 1994, https://doi.org/10.1016/0166-8641(94)00014-X
[DekimpeDere2014]
Dekimpe, Karel and Der{\'e, Expanding maps and non-trivial self-covers on infra-nilmanifolds, 2014, https://arxiv.org/abs/1407.8106
[BondarenkoZashkolny2024]
Bondarenko, Ievgen and Zashkolny, Dmytro, Virtual endomorphisms of the group $pg$, Researches in Mathematics, 2024
[Gardam2023]
Gardam, Giles, Non-trivial units of complex group rings, 2023, https://arxiv.org/abs/2312.05240
[NekrashevychSidki2004]
Nekrashevych, Volodymyr and Sidki, Said, Automorphisms of the binary tree: state-closed subgroups and dynamics of $1/2$-endomorphisms, 2004
7
.

Symbols