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Combinatorial
and Geometric
Group Theory
Dortmund and Ottawa-Montreal
Conferences
Oleg Bogopolski
Inna Bumagin
Olga Kharlampovich
Enric Ventura
Editors
Birkhäuser
Editors:
Oleg Bogopolski
Mathematisches Institut der
Heinrich-Heine-Universität Düsseldorf
Universitätsstr. 1
40225 Düsseldorf
Germany
e-mail:
Olga Kharlampovich
Department of Mathematics and Statistics
McGill University
805 Sherbrooke St., West
Montreal, Quebec, H3A 2K6
Canada
e-mail:
Inna Bumagin
School of Mathematics and Statistics
Carleton University
1125 Colonel By Drive
Ottawa, Ontario, K1S 5B6
Canada
e-mail:
Enric Ventura
Departament de Matematica Aplicada III
EPSEM – Universitat Politècnica de Catalunya
Av. Bases de Manresa 61–73
08242 Manresa, Barcelona
Spain
e-mail:
2000 Mathematics Subject Classification 20A, 20E, 20F, 20H, 20M, 20P, 03B, 03D, 05C, 08A, 51F,
57M, 57S, 60B, 68Q
Library of Congress Control Number: 2010926413
Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists
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ISBN 978-3-7643-9910-8
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Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
O. Bogopolski and R. Vikentiev
Subgroups of Small Index in Aut(Fn ) and
Kazhdan’s Property (T) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
P. Brinkmann
Dynamics of Free Group Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
V. Diekert, A.J. Duncan and A.G. Myasnikov
Geodesic Rewriting Systems and Pregroups . . . . . . . . . . . . . . . . . . . . . . . . . .
55
E. Frenkel, A.G. Myasnikov and V.N. Remeslennikov
Regular Sets and Counting in Free Groups . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
D. Gon¸calves and P. Wong
Twisted Conjugacy for Virtually Cyclic Groups and
Crystallographic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
M. Hock and B. Tsaban
Solving Random Equations in Garside Groups Using
Length Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
A. Juh´
asz
An Application of Word Combinatorics to Decision Problems
in Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
171
O. Kharlampovich and A.G. Myasnikov
Equations and Fully Residually Free Groups . . . . . . . . . . . . . . . . . . . . . . . . . 203
M. Lustig
The FN -action on the Product of the Two Limit Trees
for an Iwip Automorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
F. Matucci
Mather Invariants in Groups of Piecewise-linear
Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
vi
Contents
P.V. Morar and A.N. Shevlyakov
Algebraic Geometry over the Additive Monoid of Natural Numbers:
Systems of Coefficient Free Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
D. Savchuk
Some Graphs Related to Thompson’s Group F . . . . . . . . . . . . . . . . . . . . . .
279
R. Weidmann
Generating Tuples of Virtually Free Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 297
R. Zarzycki
Limits of Thompson’s Group F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
Combinatorial and Geometric Group Theory
Trends in Mathematics, vii–viii
c 2010 Springer Basel AG
Preface
We are pleased to present the book “Geometric Group Theory, Dortmund and
Carleton Conferences”, a selection of the best research articles from two strongly
related 2007 international conferences:
• “Combinatorial and Geometric Group Theory with Applications” (GAGTA),
the University of Dortmund (Germany) from August 27th to 31st;
• “Fields Workshop in Asymptotic Group Theory and Cryptography”, Carleton University (Ottawa, Canada) from December 14th to 16th, followed by
“Workshop on Actions on Trees, Non-Archimedian Words, and Asymptotic
Cones”, Saint Sauveur (Montreal) from December 17th to 21st.
The book contains a selection of refereed papers on Combinatorial and Geometric Group Theory. The breadth of topics included will assure the interest of all
specialists and researchers in this area of mathematics; they will also prove to
be valuable to graduate students and mathematicians in other areas who wish to
explore deeper into this exciting and very active field of research.
The articles largely fall into five categories:
• equations and algebraic geometry over groups; Tarski problems,
• algorithmic problems in groups,
• groups of automorphisms of non-abelian free groups,
• groups of transformations of the unit interval and Thompson’s group F ,
• questions motivated by group-based cryptography.
Readers interested in the first topic may choose to look first at the excellent expository paper by O. Kharlampovich and A.G. Myasnikov. Here, the authors explain
their multifaceted techniques (part of them on algebraic geometry over groups) for
solving two of Tarski’s famous problems on elementary theories of free groups. The
paper of P. Morar and A. Shevlyakov initiates investigations of algebraic geometry
over some intriquing classes of monoids.
One can also learn a lot about dynamics of automorphisms of free groups
via train tracks and actions on trees, by reading the thought-provoking papers
of P. Brinkmann and M. Lustig. In a similar direction, R. Weidmann shows how
Makanin-Razborov diagrams and Stallings foldings can be used to solve the rank
problem for virtually free groups.
viii
Preface
The paper by O. Bogopolski and A. Vikentiev describes some particularly
useful finite index subgroups of the automorphism group of a finitely generated free
group. One of their uses may be to attack the problem on the Kazhdan property
(T ) for these groups. The paper of A. Juhasz contains a solution of the difficult
membership problem in a subclass of one-relator groups.
Papers of F. Matucci, D. Savchuk and R. Zarzycki will attract the attention
of those who want to know more about groups of transformations of the unit
interval [0, 1], in particular about the famous Thompson’s group F and its limit
properties.
The paper by A.J. Duncan, V. Dieckert and A.G. Myasnikov contains a very
thorough survey on rewriting systems with new issues on infinite rewriting systems.
The paper by L. Frenkel, A.G. Myasnikov and V.N. Remeslennikov is devoted to
the problem of how to measure some subsets in free groups by using random walks.
The results of this paper may be used for designing algorithms that run fast on
almost all inputs. This paper as well as the paper by M. Hock and B. Tsaban are
highly recommended to specialists in cryptography.
Finally, the paper by D. Goncalves and P. Wong is devoted to the twisted
conjugacy in 2-dimensional crystallographic groups.
We are very grateful to the organizations that supported these two conferences:
• The conference in Dortmund was organized by O. Bogopolski, M.-T. Bochnig,
G. Rosenberger, V. Shpilrain and E. Ventura. This conference was financially
supported by DAAD (Deutscher Akademischer Austauschdienst), by DFG
(Deutsche Forschungsgemeinschaft), and by the Universită
at Dortmund. The
URL address for its homepage is
/>ã The workshops in Canada were co-organized by I. Bumagin, O. Kharlampovich and A.G. Myasnikov. The workshops could not have been held without
the generous support of the Fields Institute. The organizers also gratefully
acknowledge the financial support provided by the Faculty of Science of Carleton University and by McGill University. More information about the workshops can be found at the URL
/>scientific/07-08/asympotic/index.html
Finally, we wish to thank the contributors to this volume, and the anonymous
referees who ensured the high quality of its contents. Our thanks also go to Thomas
Hemping at Birkhă
auser for his assistance in the typsetting and preparation of this
volume. Without these joint efforts, this book would never have appeared.
The editors,
O.
I.
O.
E.
Bogopolski,
Bumagin,
Kharlampovich,
Ventura
Combinatorial and Geometric Group Theory
Trends in Mathematics, 1–17
c 2010 Springer Basel AG
Subgroups of Small Index in Aut(Fn)
and Kazhdan’s Property (T)
O. Bogopolski and R. Vikentiev
Abstract. We introduce a series of interesting subgroups of finite index in
Aut(Fn ). One of them has index 42 in Aut(F3 ) and infinite abelianization.
This implies that Aut(F3 ) does not have Kazhdan’s property (T); see [15]
and [5] for other proofs. We prove also that every subgroup of finite index
in Aut(Fn ), n 3, which contains the subgroup of IA-automorphisms has a
finite abelianization.
We introduce a subgroup K(n) of finite index in Aut(Fn ) and show,
that its abelianization is infinite for n = 3, and it is finite for n 4. We ask,
whether the abelianization of its commutator subgroup K(n) is infinite for
n 4. If so, then Aut(Fn ) would not have Kazhdan’s property (T) for n 4.
Mathematics Subject Classification (2000). 20F28, 20E05, 20E15.
Keywords. Automorphisms, free groups, Kazhdan’s property (T), congruence
subgroups.
1. Definitions, problems and motivations
In the mid 60’s, D. Kazhdan defined his property (T) for locally compact groups
and used it as a tool to demonstrate that many lattices in these groups are finitely
generated [8]. Later this property found various surprising applications, in particular, in the first explicit construction of expander graphs by G. Margulis [14], see
also the book of A. Lubotzky [9] and the paper of A. Lubotzky and I. Pak [10]. We
recommend to the reader the very informative book of B. Bekka, P. de la Harpe
and A. Valette [2] and the lecture of Y. Shalom [21] on the property (T) and its
applications.
There are several equivalent definitions of the property (T) for topological
groups. Below we give one of them in case, where the group is finitely generated.
We will assume that the group is endowed with the discrete topology.
Let G be a finitely generated group and π : G → U(H) a unitary representation of G on a Hilbert space H. If S is a finite generating set of G and > 0, then
2
O. Bogopolski and R. Vikentiev
a unit vector v in H is called an (S, )-invariant vector if ||π(s)(v) − v|| <
s ∈ S.
for all
Definition 1.1. A finitely generated group G has the property (T) iff for every
finite generating subset S ⊆ G there exists some > 0, such that the following is
satisfied:
For any unitary representation π : G → U(H) on a Hilbert space H, the
existence of an (S, )-invariant vector implies the existence of a non-zero
π(G)-invariant vector.
Any such is called a Kazhdan constant for G with respect to S. It is easy
to show, that in this definition the words “for every finite generating set” can be
replaced by “for some finite generating set”.
Definition 1.2. A finitely generated group G has property (FH) if any action of G
by affine isometries on a Hilbert space has a fixed point.
By a theorem of Delorme–Guichardet (see Theorem 2.12.4 in [2]), the properties (T) and (FH) are equivalent for finitely generated groups. There are strong
consequences on several types of actions: for a group with the property (FH), any
isometric action on a tree has a fixed point or a fixed edge (this is the property
(FA) of Serre), any isometric action on a real or complex hyperbolic space has a
fixed point.
Definition 1.3. A group G has Serre’s property (FA) if acting simplicially and
without inversions of edges on any simplicial tree, G has a global fixed point.
If G is finitely generated, this property can be reformulated in purely algebraic terms.
Theorem 1.4. (J.-P. Serre; 1974). A finitely generated group G has the property
(FA) if and only if the following two statements hold:
(1) G is not a nontrivial amalgamated product, that is G
and C = B.
(2) G does not have a quotient isomorphic to Z.
A ∗C B with C = A
Theorem 1.5. (Y. Watatani; 1982). Let G be a countable group. If G has the
property (T) of Kazhdan, then it has the property (FA) of Serre.
Let Fn be the free group of rank n with basis x1 , . . . , xn . In this paper we will
concentrate on the group Aut(Fn ), the automorphism group of Fn . There is the
canonical epimorphism Φ : Aut(Fn ) → GLn (Z), which sends an automorphism
α ∈ Aut(Fn ) to the matrix, whose ij-entry equals to the sum of exponents of
xj in α(xi ), i, j = 1, . . . , n. The full preimage of SLn (Z) is called the special
automorphism group of Fn and is denoted by SAut(Fn ). The kernel of Φ is denoted
by IA(Fn ).
Subgroups of Small Index in Aut(Fn )
3
In the following table we summarize known facts on the (T) and (FA) properties for SLn (Z) and SAut(Fn ), n 3.
(T)
SLn (Z), n
3
SAut (F3 ),
SAut (Fn ), n
4,
⇒ (FA)
+
(Kazhdan, 1967)
+
(Serre, 1974)
−
(McCool, 1989)
+
(Bogopolski, 1987)
?
+
(Bogopolski, 1987)
Remark 1.6. The property (T) is preserved under taking subgroups of finite index,
the property (FA) is not preserved. Both properties are preserved under taking
overgroups of finite index. Groups with the property (T) have no subgroups of
finite index, which can be mapped onto Z.
Problems 1.7.
1) Does the group Aut(Fn ), n 4, have the property (T)?
2) Does every finite index subgroup of Aut(Fn ), n 4, have the property (FA)?
3) Characterize (in terms of actions on trees or algebraically) those finitely generated groups, whose subgroups of finite index have the (FA) property.
In [23], K. Vogtmann formulated the Out-versions of the problems 1) and 2)
(see Problems 14 and 15 there). The first problem is also formulated as Problem
(7.1) in the list of open questions in [2].
Due to A. Lubotzky and I. Pak [10], the presence of the Kazhdan property
for Aut(Fn ) would imply a very elegant construction of an infinite series of εexpanders. The later have applications to theoretical computer science, design of
robust computer networks, and the theory of error-correcting codes [7].
Definition 1.8. Let ε be a positive real number. A finite d-regular graph Γ is called
an ε-expander, if for every subset of vertices B ⊂ Γ0 with |B| |Γ0 |/2, we have
|∂B| ε|B|, where
∂B = {v ∈ Γ0 | v ∈
/ B, but v is adjacent to a vertex in B}.
Definition 1.9 (Graph Γn (G)). Fix a natural number n and a finite group G,
which can be generated by n elements. The vertices of the graph Γn (G) are all
tuples (g) = (g1 , . . . , gn ) such that g1 , . . . , gn = G. Two tuples (g) and (g ) are
connected by an edge if (g ) can be obtained from (g) by applying one of the
following replacement operations:
±
: (g1 , . . . , gi , . . . , gn ) → (g1 , . . . , gi · gj±1 , . . . , gn ),
Ri,j
±1
L±
i,j : (g1 , . . . , gi , . . . , gn ) → (g1 , . . . , gj · gi , . . . , gn ),
where i = j.
4
O. Bogopolski and R. Vikentiev
Clearly, the graph Γn (G) is 4n(n − 1)-regular.
The following theorem is a partial case of a more general Theorem 3.1 in [10].
It explains why establishing the Kazhdan property (T) for Aut(Fn ) is important.
Theorem 1.10 (A. Lubotzky and I. Pak; 2001). If the group Aut(Fn ) (or equivalently SAut(Fn )) has the property (T), then there exists an ε > 0, such that every
connected component of the graph Γn (G) is an ε-expander for any n-generated
finite group G.
The group Aut(F2 ) does not have the property (T), since it can be represented
as a nontrivial amalgamated product. (The uniqueness of such representation up
to conjugacy was shown by O. Bogopolski in [3]). The following two theorems
imply that Aut(F3 ) does not have the property (T). In this paper we suggest some
ideas for studying this problem in rank n 4.
Theorem 1.11 (J. McCool; 1989). There is a subgroup of finite index in Out(F3 ),
which can be approximated by torsion-free nilpotent groups. In particular, this subgroup can be mapped onto Z. Therefore Out(F3 ) and Aut(F3 ) do not have the
Kazhdan property (T).
Theorem 1.12 (F. Grunewald and A. Lubotzky; 2006). There exists a subgroup of
index 168 in Aut(F3 ) which can be mapped onto F2 . In particular, Aut(F3 ) has no
Kazhdan’s property (T).
The proof of this theorem has existed for a long time in a folklore form. We
know variants of the proof from private talks with A. Casson (2000) and M. Bridson
(2004). To our best knowledge, the first written proof (based on the same idea)
appeared in the paper of F. Grunewald and A. Lubotzky [5], see Corollary 1.3 there.
This corollary is deduced there from a more general Theorem 1.1 on representations
of Aut(Fn ).
The structure of the paper is as follows. In Section 2 we give a short exposition of the proof of Theorem 1.12. Some elements of this proof motivated us to
further constructions. In Section 3 we introduce some useful automorphisms. In
Section 4 we prove that any finite index subgroup of Aut(Fn ) containing IA(Fn )
has finite abelianization (Theorem 4.1). Thus, to construct a finite index subgroup
in Aut(Fn ) with infinite abelianization, one should avoid IA-automorphisms.
In Section 5 we introduce and investigate congruence subgroups Cong(n, m)
and SCong(n, m) in Aut(Fn ) and SAut(Fn ) respectively. Both congruence subgroups contain IA(Fn ).
In Section 6 we define a subgroup K(n) of index 2 in SCong(n, 2), so that
it does not contain IA(Fn ). In Section 7 we show that K(3) has infinite abelianization. This gives an alternative proof of Theorem 1.12. Further we construct a
series of overgroups of K(3):
Aut(F3 )
C(3)
42
B(3)
2
A(3)
2
K(3).
2
The largest one, C(3), has index 42 in Aut(F3 ) and infinite abelianization. We
conjecture, that this is the minimal possible index for a subgroup in Aut(F3 )
Subgroups of Small Index in Aut(Fn )
5
with infinite abelianization. We compute the abelianizations of these subgroups
explicitly:
K(3)/K(3) ∼
= Z14 × Z × Z,
2
A(3)/A(3) ∼
= Z72 × Z × Z,
B(3)/B(3) ∼
= Z4 × Z,
2
C(3)/C(3) ∼
= Z32 × Z4 × Z.
Here we use the notation Zkn = Z/nZ × · · · × Z/nZ.
k times
In Section 8 we prove that if n 4, then the abelianization of K(n) is a finite
2-group. In particular, the abelianization of K(4) is Z38
2 . We would like to know,
what is the abelianization of the commutator subgroup K(n) for n
4. If it is
infinite, the group Aut(Fn ) does not have Kazhdan’s property (T) for n 4.
In this paper we use the following notation for the commutator: [x, y] =
xyx−1 y −1 .
2. A sketch of the proof of F. Grunewald and A. Lubotzky
that Aut(F3) has no Kazhdan’s property (T)
Let F3 = F (a, b, c) be the free group on free generators a, b, c. There exist exactly 7
epimorphisms F (a, b, c) → Z2 . Therefore there exist exactly 7 subgroups of index
(1)
(7)
2 in F3 . Denote them by F5 , . . . , F5 . Clearly, every such subgroup has rank 5.
(1)
We will work with one of them, F5 = a, b, c2 , c−1 ac, c−1 bc . It is easy to check
that Aut(F3 ) acts transitively on the set of these subgroups. Therefore the index
(1)
of St(F5 ) in Aut(F3 ) is 7, where
(1)
(1)
(1)
St(F5 ) = α ∈ Aut(F3 ) | α(F5 ) = F5
(1)
.
(1)
Clearly, the restriction map Res : St(F5 ) → Aut(F5 ), α → α|F (1) , is an
5
embedding. Now we introduce an important inner automorphism τ : x → c−1 xc,
x ∈ F3 .
Lemma 2.1.
(1)
1) τ ∈ St(F5 ),
(1)
2) τ |F (1) ∈
/ Inn(F5 ),
5
3) (τ −1 ϕ−1 τ ϕ)|F (1) ∈ Inn(F5 ) for every ϕ ∈ St(F5 ).
(1)
(1)
5
Proof. The first two statements are straightforward. We prove the third one. Let
(1)
(1)
x ∈ F5 and ϕ ∈ St(F5 ). An easy computation shows that xτ −1 ϕ−1 τ ϕ =
(1)
ϕ(c−1 )cxc−1 ϕ(c). Thus we need to show that c−1 ϕ(c) ∈ F5 . The last is evident,
(1)
(1)
since F5 and hence the second coset cF5 are ϕ-invariant.
6
O. Bogopolski and R. Vikentiev
Consider the composition of two homomorphisms
Res
(1)
(1)
Ψ : St(F5 ) −→ Aut(F5 ) −→ GL5 (Z),
(1)
where the second homomorphism sends an automorphism of F5 to the automor(1)
phism induced on the abelianization of F5 (we identify the last automorphism
(1)
with the corresponding matrix, using the prescribed basis of F5 ).
One can easily compute, that
⎞
⎛
0 0 0 1 0
⎜ 0 0 0 0 1 ⎟
⎟
⎜
⎟
Ψ(τ ) = ⎜
⎜ 0 0 1 0 0 ⎟.
⎝ 1 0 0 0 0 ⎠
0 1 0 0 0
Since (Ψ(τ ))2 = Id, we have Z5 ⊃ V+ ⊕ V− , where
V− = Ker(Ψ(τ ) − Id).
V+ = Ker(Ψ(τ ) + Id),
The Z-submodule V+ has the basis {(1, 0, 0, −1, 0), (0, 1, 0, 0, −1)}.
(1)
By Lemma 2.1.3), the matrices Ψ(τ ) and Ψ(ϕ) commute for all ϕ ∈ St(F5 ).
(1)
Hence the submodule V+ is Ψ(ϕ)-invariant for every ϕ ∈ St(F5 ). Thus, there is
the natural homomorphism
(1)
θ : St(F5 ) → GL(V+ ) ∼
= GL2 (Z),
θ(ϕ) = Ψ(ϕ)
V+
.
This homomorphism is onto, since the automorphisms ϕ1 : a → a, b → ba, c → c
(1)
and ϕ2 : a → b, b → a, c → c belong to St(F5 ) and are mapped onto the matrices
1 0
1 1
0 1
1 0
,
,
which generate GL2 (Z)
Notice that GL2 (Z) ∼
= D4 ∗D2 D6 , where Dm denotes the dihedral group
of order 2m (see [4] for example). Therefore there exists an epimorphism μ :
GL2 (Z) → D12 . The kernel of μ is a free group of rank 2, we denote it by F2 . Thus
we have the following chain of embeddings and epimorphisms:
Aut(F3 )
(1)
θ
μ
St(F5 ) → GL2 (Z) → D12 .
(1)
Let H = Ker(θμ). Then H has index 24 in St(F5 ). Hence H has index 168
in Aut(F3 ). Moreover, θ(H) = Ker(μ) = F2 . In particular, H can be homomorphically mapped onto Z. Hence Aut(F3 ) does not have the Kazhdan property (T).
Remark 2.2. The group H is not normal in Aut(F3 ).
The above construction cannot be generalized for Aut(Fn ), where n
4,
since in this case GLn−1 (Z) (contrary to GL2 (Z)) does not contain a subgroup of
finite index with infinite abelianization.
Subgroups of Small Index in Aut(Fn )
7
3. Some notations and useful automorphisms
Let Fn be the free group on free generators x1 , x2 , . . . , xn . First we define some
automorphisms of Fn . We will write the image of xi only if it differs from xi .
1) For any i, j, k ∈ {1, 2, . . . , n}, where k = i, j, we define the automorphism
−1
αijk : xi → xi · x−1
j xk xj xk .
In particular,
αiik : xi → x−1
k xi xk .
Note that αijk = α−1
ikj for distinct i, j, k. We say that the automorphism αijk
is of the first kind if i, j, k are distinct, and of the second kind if i = j.
2) For any i, j ∈ {1, 2, . . . , n}, where i = j, we define the automorphism
Eij : xi → xi xj .
3) For any i ∈ {1, 2, . . . , n} we define the automorphism
Ni : xi → x−1
i .
We denote Nij = Ni Nj for i = j.
The kernel of the canonical epimorphism Aut(Fn ) → GLn (Z) is denoted by
IA(Fn ). It is known, that IA(F2 ) = Inn(F2 ) and IA(Fn ) is strictly larger than
Inn(Fn ) for n 3. J. Nielsen for n 3 [17] and W. Magnus for all n [12] proved
that IA(Fn ) is generated by all automorphisms αijk (see also [11]).
4. Finite index subgroups of Aut(Fn) containing IA(Fn)
Theorem 4.1. Let n
3. Any subgroup of finite index in Aut(Fn ), containing
IA(Fn ), has a finite abelianization.
To prove this theorem we need to introduce more automorphisms of Aut(Fn )
and to formulate a theorem of B. Sury and T.N. Venkataramana (see below) on
generators of congruence subgroups of SLn (Z).
4) For any i ∈ {1, 2, . . . , n} we define the automorphism
−1
Ti : xi → x−1
i , xi+1 → xi+1 xi .
5) For any i, j ∈ {1, 2, . . . , n}, where i = j, we define the automorphism
Tij : xi → xj , xj → x−1
i .
Denote
T = {Tk | k = 1, . . . , n − 1} ∪ {Tij | i, j = 1, 2, . . . , n; i = j} ∪ {I},
where I is the identity automorphism of Fn .
Let ¯ : Aut(Fn ) → GLn (Z) be the canonical epimorphism. For any α ∈
Aut(Fn ) we denote by α its canonical image in GLn (Z).
8
O. Bogopolski and R. Vikentiev
Let m be a natural number. The kernel of the canonical epimorphism
SLn (Z) → SLn (Zm ) is denoted by SLn (Z, mZ) and is called the congruence subgroup of SLn (Z) modulo m. Of course, SLn (Z, mZ) is normal and has a finite index
in SLn (Z). The following theorem is called the congruence subgroup theorem for
SLn (Z). It was proved by H. Bass, M. Lazard and J.-P. Serre [1] and independently
by J. Mennicke [16].
Theorem 4.2. Let n
3 be a natural number. Any subgroup of finite index in
SLn (Z) contains a congruence subgroup SLn (Z, mZ) for some m.
Theorem 4.3. (B. Sury and T.N. Venkataramana; 1994). Let n 3, m 2. The
congruence subgroup SLn (Z, mZ) is generated by the following set of matrices
{(α)(E ij )m (α)−1 | α ∈ T ; i, j ∈ {1, 2, . . . , n}; i = j}.
Corollary 4.4. The group SLn (Z, 2Z), n
X=
2
{E ij , N ij
3, is generated by the set
| i, j ∈ {1, 2, . . . , n}; i = j}.
2
Proof. By Theorem 4.3, SLn (Z, 2Z) is the normal closure of all E ij in SLn (Z).
Since SLn (Z) is generated by all transvections E pq , it is sufficient to verify that
for every x ∈ X and
reader.
−
∈ {−1, 1} holds E pq xE pq ∈ X . We leave this to the
Proof of Theorem 4.1. Let G be a subgroup of finite index in Aut(Fn ) containing
IA(Fn ). We show that G/G is finite. Let G be the image of G is GLn (Z). The index
of G ∩ SLn (Z) in SLn (Z) is finite. Hence, by the congruence subgroup theorem,
G. Let H be the full preimage
there exists an m
2, such that SLn (Z, mZ)
of SLn (Z, mZ) in G. Since the index of H in G is finite, it is sufficient to show
that H/H is finite. Since H contains IA(Fn ), the results of Nielsen–Magnus and
Sury–Venkataramana imply that H is generated by the union of two sets:
{αijk | i, j, k ∈ {1, 2, . . . , n}; k = i, j},
m −1
α | α ∈ T ; i, j ∈ {1, 2, . . . , n}; i = j}.
{αEij
It is sufficient to prove that the mth power of each of these generators lies in
[H, H]. But this follows from the following formulas:
m
1) [αiij , Ejk
] = αm
iik for distinct i, j, k ∈ {1, 2, . . . , n};
m
m−1
αikj αm−1
≡ αm
2) [αiij , Eik ] = (αikj α−1
ikj mod[IA(Fn ), IA(Fn )] for disjjk )
jjk
tinct i, j, k ∈ {1, 2, . . . , n};
m2 −(s+1)m
m2 −(s+1)m
m−2 s
m2
m
m
=
][Eik
, αs+1
3) Eik
iij ] [Eij , Ejk ].
s=0 [αiij , Eik
5. Congruence subgroups SCong(n, k) in SAut(Fn)
Let G be a group and H be a normal subgroup in G. We denote
Aut(G; H) = {ϕ ∈ Aut(G) | ϕ(H) = H}.
Subgroups of Small Index in Aut(Fn )
9
and
IA(G; H) = {ϕ ∈ Aut(G) | ∀g ∈ G : ϕ(gH) = gH}.
Equivalently
IA(G; H) = {ϕ ∈ Aut(G) | ∀g ∈ G ∃xg ∈ H : ϕ(g) = g · xg }.
(1)
Clearly, IA(G; H) is normal in Aut(G; H) and the corresponding factor group
is naturally embeddable into Aut(G/H). In particular, if |G : H| = 2, then we have
IA(G; H) = Aut(G; H).
Proposition 5.1. Let {Hi | i ∈ I} be a set of normal subgroups of a group G. Then
IA G; ∩ Hi = ∩ IA G; Hi .
i∈I
i∈I
Proof. The proof is straightforward from description (1).
Now we return to automorphisms of Fn = F (x1 , . . . , xn ). Let k
natural number. Consider the standard epimorphisms
π
2 be a
ε
Fn −→ Zn −→ Znk .
They induce the epimorphisms
π
ε
Aut(Fn ) −→ Aut(Zn ) −→ Aut(Znk ).
Using usual identifications we may write
π
ε
Aut(Fn ) −→ GLn (Z) −→ GLn (Zk ).
We denote
Cong(n; k) = Ker(π ),
SCong(n; k) = Cong(n; k) ∩ SAut(Fn )
and call these groups the congruence subgroups of Aut(Fn ) and of SAut(Fn ) modulo k, respectively. In this paper we will work only with congruence subgroups
modulo 2.
Let {σi | i = 1, . . . , 2n − 1} be the set of all epimorphisms Fn → Z2 . The
(i)
kernel of σi is a free group of rank 2n − 1; we denote it by F2n−1 . We fix σ1 by
(1)
the rule σ1 (x1 ) = · · · = σ1 (xn−1 ) = 0 and σ1 (xn ) = 1. Thus, F2n−1 has the basis
2
−1
−1
{x1 , . . . , xn−1 , xn , xn x1 xn , . . . , xn xn−1 xn }. For brevity we denote
(i)
(i)
St(F2n−1 ) = Aut(Fn , F2n−1 ).
Proposition 5.2. For n
3 holds:
1) Aut(Fn )/Cong(n, 2) ∼
= GLn (Z2 ),
2) |Cong(n, 2) : SCong(n, 2)| = 2,
3) Cong(n, 2) =
2n −1
i=1
(i)
St(F2n−1 ),
10
O. Bogopolski and R. Vikentiev
4) SCong(n, 2) is generated by the set
X = {αijk | i, j, k ∈ {1, 2, . . . , n}, k = i, j}
2
, Nij | i, j ∈ {1, 2, . . . , n}, i = j}.
{Eij
5) The abelianization of SCong(n, 2) is a finite 2-group.
Proof. 1) This statement follows from the definition of the congruence subgroup.
2) We define the automorphism ϕ ∈ Aut(Fn ) by the rule x1 → x1 x22 , x2 →
x1 x22 x1 x2 , xi → xi for i = 1, . . . , n. It is easy to see, that ϕ ∈ Cong(n, 2) \
SCong(n, 2).
3) This statement follows from the chain of identities (with k = 2):
2n −1
Ker(π ε) = IA Fn ; Ker(πε) =
2n −1
IA Fn ; Ker(σi ) =
i=1
Aut Fn ; Ker(σi ) .
i=1
The first identity is evident, the second one follows from Proposition 5.1 and the
fact that Ker(πε) =
2n −1
Ker(σi ). The third identity follows from the fact that
i=1
|Fn : Ker (σi )| = 2.
4) This statement follows from Corollary 4.4 and Nielsen–Magnus result.
5) The abelianization of SCong(n, 2) is finite by Theorem 4.1. The following
computations (where i, j, k are distinct) show, that it is a finite 2-group:
2
[αiij , Ejk
] = α2iik ,
2
2
[αiij , Eik
] = αikj α−1
jjk αikj αjjk ≡ αikj mod IA(Fn ) ,
2
4
[Eij
, Njk ] = Eij
,
Nij2 = 1.
Remark 5.3. Using first Johnson homomorphism and some homological methods,
T. Satoh proved in [19], that for n 2 and k 2 holds
Cong(n, k) ∼
= IA(Fn ) ⊗Z Zk ⊕ Γ(n, k) ,
where Γ(n, k) is the congruence subgroup of GLn (Z) modulo k.
6. A subgroup K(n) of index 2 in SCong(n, 2)
We have the following chain of canonical embeddings and epimorphisms.
θ1
(1)
θ2
θ
SCong(n, 2) → St(F2n−1 ) → Aut(F2n−1 ) →3 GL2n−1 (Z).
Here θ1 is the embedding due to Proposition 5.2.3, θ2 is the homomorphism,
(1)
(1)
which sends every automorphism α of Fn with α(F2n−1 ) = F2n−1 to its restriction
(1)
on F2n−1 . In fact, θ2 is injective. The epimorphism θ3 is standard.
Now we set θ = θ1 θ2 θ3 and define the subgroup
K(n) = θ−1 SL2n−1 (Z) .
Subgroups of Small Index in Aut(Fn )
Proposition 6.1. For n
1)
2)
3)
4)
11
3 holds
|SCong(n, 2) : K(n)| = 2,
|IA(Fn ) : IA(Fn ) ∩ K(n)| = 2,
|Inn(Fn ) : Inn(Fn ) ∩ K(n)| is 1 if n is odd, and is 2 if n is even,
K(n) is generated by the set
Y = X \ {αiin , Nin | i = 1, 2, . . . , n − 1} ∪ {N1n αiin | i = 1, 2, . . . , n − 1},
where
X = {αijk | i, j, k ∈ {1, 2, . . . , n}, k = i, j}
2
, Nij | i, j ∈ {1, 2, . . . , n}, i = j}
{Eij
is the generating set for SCong(n, 2) defined in Proposition 5.2.
Proof. 1) Clearly, |SCong(n, 2) : K(n)|
2. One can check that
N1n ∈ SCong(n, 2) \ K(n).
Therefore this index is indeed 2.
2) Since IA(Fn )
SCong(n, 2), we have by 1), that |IA(Fn ) : IA(Fn ) ∩
K(n)|
2. One can verify, that α11n ∈ IA(Fn ) \ K(n). Therefore this index is
indeed 2.
3) By 2), this index does not exceed 2. For x ∈ Fn let x
ˆ be the conjugation
of Fn by x, i.e., x
ˆ(y) = x−1 yx for y ∈ Fn . To prove the statement, it is sufficient
to check that x
ˆ1 , . . . , xˆn−1 ∈ K(n) and that x
ˆn ∈ K(n) if and only if n is odd.
4) We take {1, N1n } as the set of representatives of the cosets of K(n) in
SCong(n, 2). For g ∈ SCong(n, 2), we denote by g the representative of the coset
K(n)g. One can easily check, that for g ∈ X holds
g=
N1n , if g ∈ {Nin , αiin | i = 1, . . . , n − 1}
1,
if g ∈ X \ {Nin , αiin | i = 1, . . . , n − 1}.
By the Reidemeister–Schreier method, K(n) is generated by the elements
−1
N1n xN1n x , where x runs through X. We show, that these elements can be
expressed as products of elements of Y ±1 . Consider the following cases.
I. Let x = αiin , where i ∈ {1, . . . , n − 1}.
−1
Then N1n xN1n x
= N1n αiin ∈ Y .
II. Let x = αijk , where i, j, k ∈ {1, . . . , n} are distinct or x = αiik with k = n (the
case x = αiik with k = n was considered above).
−1
−1
Then N1n xN1n x = N1n αijk N1n
.
We consider several cases and rewrite this element as a product of elements
of Y ±1 . We will use the fact that αijk = α−1
ikj if i, j, k are distinct.
Case 1. {1, n} ⊆ {i, j, k}.
Subcase 1.1. αijk is of the first kind, i.e., i, j, k are different.
12
O. Bogopolski and R. Vikentiev
a) i = 1. Then n ∈ {j, k} and we may assume that n = j. Then
−1
−2
−1
2
N1n α1nk N1n
= N1k α−1
11k E1n α1kn α11k E1n N1k .
b) i = n. Then 1 ∈ {j, k} and we may assume that j = 1. Then
−1
−1 −1
−1
= α−1
N1n αn1k N1n
n1k αnnk αnn1 αnnk αnn1 αkk1 αn1k αkk1 αn1k .
c) i = 1, n. Then {1, n} = {j, k} and we may assume that j = 1 and k = n.
Then
−1
−2 −1
2
N1n αi1n N1n
= Ein
αii1 αin1 Ein
αii1 .
Subcase 1.2. αijk is of the second kind, i.e., it is αiik . By our assumption in II we
have k = n. Therefore i = n and k = 1, and we have
−1
N1n αnn1 N1n
= α−1
nn1 .
Case 2. n ∈ {i, j, k}, 1 ∈
/ {i, j, k}. Then we choose t ∈ {i, j, k} \ {n} and write
−1
−1
−1
= N1t (Ntn αijk Ntn
)N1t
.
N1n αijk N1n
The expression in the brackets can be computed as in Case 1 (by replacing 1 by t).
Case 3. n ∈
/ {i, j, k}, 1 ∈ {i, j, k}.
Subcase 3.1. αijk is of the first kind, i.e., i, j, k are different.
a) i = 1. Then
−1
−1
N1n α1jk N1n
= α1kj · α11k α11j α−1
11k α11j .
b) j = 1. Then
−1
= α−1
N1n αi1k N1n
ii1 αik1 αii1 .
c) k = 1. This case reduces to Case b), since αijk = α−1
ikj .
Subcase 3.2. αijk is of the second kind, i.e., it is αiik .
a) i = 1. Then
−1
= α11k .
N1n α11k N1n
b) k = 1. Then
−1
N1n αii1 N1n
= α−1
ii1 .
Case 4. {1, n} ∩ {i, j, k} = ∅.
−1
N1n αijk N1n
= αijk .
Subgroups of Small Index in Aut(Fn )
2
.
III. Let x = Eij
13
−1
−1
2
Then N1n xN1n x = N1n Eij
N1n
. The following formulas show, that this
element belongs to Y .
−1
−2
2
N1n
= α2nnj Enj
N1n Enj
(j = 1, n),
−1
−2
2
N1n E1j
N1n
= α211j E1j
(j = 1, n),
−1
2
N1n
N1n Ein
−1
2
N1n Ei1
N1n
−1
2
N1n En1
N1n
−1
2
N1n E1n
N1n
IV. Let x = Nij .
If i, j = n, then N1n xN1n x
−1
N1n xN1n x
=
=
=
=
−2
Ein
(i = 1, n),
−2
Ei1 (i = 1, n),
2
α−2
nn1 En1 ,
−2 −1
N12 E1n
N12 .
−1
= Nij . If, say j = n, then i = n and
= N1i .
7. K(3) and some its overgroups with infinite abelianization
The congruence subgroup SCong(3, 2) has index 2 · 168 in Aut(F3 ) and contains
IA(F3 ). Therefore, by Theorem 4.1, it has a finite abelianization. Moreover, by
Proposition 5.2, this abelianization is a finite 2-group.
The subgroup K(3) has index 2 in SCong(3, 2) and does not contain IA(F3 )
by Proposition 6.1. This was an indication for us to check that the abelianization
of K(3) is infinite. In this section we also construct some overgroups of K(3),
namely A(3), B(3) and C(3), with infinite abelianizations.
In computing the abelianizations of these groups, we used their generators,
Nielsen’s presentation of Aut(F3 ) (see [18] or [13]) and the Reidemeister–Schreier
method implemented in the GAP package. By Proposition 6.1 we have the following 16
Generators of K(3):
(1) α112 , α221 , α331 , α332 , α123 , α213 , α312 ,
2
2
2
2
2
2
(2) E12
, E13
, E21
, E23
, E31
, E32
,
(3) N13 α113 , N13 α223 , N12 .
Theorem 7.1.
1) K(3) has index 4 · 168 in Aut(F3 ).
2) K(3)/K(3) ∼
= Z14
2 × Z × Z.
Corollary 7.2. The above 16 automorphisms form a minimal generating set of
2
2
, E21
have finite order in K(3)/K(3) .
K(3). All of them, except of α123 , α213 , E12
These four automorphisms have infinite order in K(3)/K(3) .
−4
−4
≡ α2123 (mod K(3) ) and E21
≡ α2213 (mod K(3) ). In particuMoreover, E12
2
2
lar, the image of the group E12 , E21 in K(3)/K(3) is isomorphic to Z × Z.
14
O. Bogopolski and R. Vikentiev
Proof. The first statement of this corollary follows straightforward from Theorem 7.1. The second statement follows from the formulas
[α221 , N12 ] = α2221 ,
[α112 , N12 ] = α2112 ,
2
] = α2332 ,
[α331 , E12
2
[α332 , E21
] = α2331 ,
−2
2
2
[α331 , E32
] = α321 α−1
112 α321 α112 ≡ α321 = α312 (mod K(3) ),
2
4
[E31
, N12 ] = E31
,
2
4
[E32
, N12 ] = E32
,
2
2
4
[α221 , E13
][E23
, N12 ] = E23
,
2
2
4
][E13
, N12 ] = E13
,
[α112 , E23
(N13 α113 )2 = 1,
(N13 α223 )2 = 1,
2
N12
= 1.
This statement and Theorem 7.1.2) imply that the image of
2
2
, E21
α123 , α213 , E12
in K(3)/K(3)
can be mapped onto Z × Z. Therefore all remaining statements of the corollary
−4
−4
will follow, if we prove the congruences E12
≡ α2123 (mod K(3) ) and E21
≡
2
α213 (mod K(3) ). The first congruence follows from the identity
−1 −2 −1 2
−4
−2 −4
2
] = α332 α−1
[α113 N23 , E12
123 α332 α112 α123 α112 E12 ≡ α123 E12 (mod K(3) )
2
and the fact that the commutator [α113 N23 , E12
] belongs to K(3) ; indeed, we have
2
α113 N23 ∈ K(3) and E12 ∈ K(3). The second congruence follows similarly if we
exchange indices 1 and 2.
Still, the index of K(3) in Aut(F3 ) is large. We will enlarge K(3) (and so
decrease the index) by adding special generators. In this way we construct the
following chain of subgroups:
Aut(F3 )
C(3)
B(3)
A(3)
K(3),
where
A(3) = K(3), E31 , E32 ,
B(3) = K(3), E31 , E32 , E21 ,
C(3) = K(3), E31 , E32 , E21 , N3 .
Subgroups of Small Index in Aut(Fn )
15
Theorem 7.3.
1) A(3) has index 168 in Aut(F3 ).
2) A(3)/A(3) ∼
= Z72 × Z × Z.
3) A(3) has the following minimal set of generators:
2
2
α112 , α221 , α123 , α213 , E13
, E23
, E31 , E32 , N12 .
Theorem 7.4.
1) B(3) has index 84 in Aut(F3 ).
2) B(3)/B(3) ∼
= Z42 × Z.
Remark. We do not know a minimal generating set of B(3).
Theorem 7.5.
1) C(3) has index 42 in Aut(F3 ).
2) C(3)/C(3) ∼
= Z32 × Z4 × Z.
3) C(3) has the following minimal set of generators:
2
α123 , E13
, E21 , E32 , N3 .
Questions 7.6. Does there exist a subgroup of Aut(F3 ) of index smaller than 42,
which can be mapped onto Z?
8. The group K(n) for n
Theorem 8.1. If n
4
4, then the abelianization of K(n) is a finite 2-group.
Proof. By Proposition 6.1, K(n) is generated by the set
Y = X \ {αiin , Nin | i = 1, 2, . . . , n − 1} ∪ {N1n αiin | i = 1, 2, . . . , n − 1},
where
X = {αijk | i, j, k ∈ {1, 2, . . . , n}, k = i, j}
2
, Nij | i, j ∈ {1, 2, . . . , n}, i = j}.
{Eij
We show, that the square of each y ∈ Y is trivial modulo K(n) .
4
1. We show that Eij
≡ 1 mod K(n) .
For j = n, this follows from the identity
−2
−4
, Njk ] = Eij
,
[Eij
where we choose k ∈
/ {i, j, n}.
For j = n, this follows from the identity
2
2
4
[αiik , Ekn
][Ein
, Nik ] = Ein
,
where we choose k ∈
/ {i, n}.
2. We show that α2iij ≡ 1 mod K(n) .
This follows from the identity
2
[αiik , Ekj
] = α2iij ,
