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ANNALES MATHÉMATIQUES
BLAISE PASCAL
Goulnara Arzhantseva & Światosław R. Gal
On approximation properties of semidirect products of groups
Volume 27, no 1 (2020), p. 125-130.
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Annales mathématiques Blaise Pascal 27, 125-130 (2020)
On approximation properties of semidirect products of groups
Goulnara Arzhantseva
Światosław R. Gal
Abstract
Let R be a class of groups closed under taking (split) extensions with finite kernel and fully residually
R–groups. We prove that R contains all (split) {finitely generated residually finite }–by–R groups. It
follows that a split extension with a finitely generated residually finite kernel and a surjunctive quotient is
surjunctive. This remained unknown even for direct products of a surjunctive group with the integers Z.
Sur les propriétés d’approximation des produits semi-directs des groupes
Résumé
Soit R une classe de groupes fermée par rapport aux extensions (scindées) avec un noyau fini et
par rapport aux groupes multi-résiduellement R. Nous montrons que R contient toutes les extensions
(scindées) de type {finiment engendré résiduellement fini}–par–R. Nous obtenons en corollaire qu’une
extension scindée avec un noyau finiment engendré résiduellement fini et un quotient surjonctif est
surjonctive. Cela restait inconnu, même pour les produits directs d’un groupe surjonctif avec les entiers Z.
The concept of approximation is one of the most fundamental in science. In nowadays
geometric group theory it refers to algebraic approximations such as residual finiteness,
to approximations in the space of marked groups such as local embeddability into a
given class of groups, e.g. [5, 13], or to metric approximations such as soficity and
hyperlinearity [2, 12].
A current intensive study of sofic and hyperlinear groups, motivated by a variety of
deep results on these wide – and still mysterious: is there a non sofic/hyperlinear group? –
classes of groups, has given rise to a number of questions on group-theoretical properties
both of these recently discovered groups and of their fundamental predecessors. One of
such a major question is whether or not a given class of groups is preserved under taking
(split or, in other words, semidirect) extensions.
Every semidirect extension of a residually finite group with a finitely generated
residually finite kernel is residually finite by an elegant result of Mal′cev [10]. A more
general is a class of groups locally embeddable into finite ones (briefly, LEF–groups).
There exist examples of semidirect extensions of LEF–groups which are not LEF [13].
The research of both authors was supported in part by the Swiss NSF, under Sinergia grant CRSI22–130435.
The research of the first author was supported in part by her ERC grant ANALYTIC no. 259527.
The research of the second author was supported in part by Polish National Science Center (NCN) grants
2012/06/A/ST1/00259 and 2017/27/B/ST1/01467.
Keywords: Residually finite groups, surjunctive and sofic groups, semidirect product.
2020 Mathematics Subject Classification: 20E26, 20E22, 20E25, 37B05, 37B10.
125
G. Arzhantseva & Ś. R. Gal
With regard to metric approximations, the only known and positive results on extensions
are on (not necessarily split) extensions by an amenable group [2, 12].
Our main result is the following general theorem which can be then applied to many
groups, including all those just mentioned, and, in particular, to surjunctive groups
(see the definition below), which came to light after Gromov’s spectacular proof of the
surjunctivity of all sofic groups [8].
Theorem 1. Assume that R is a class of groups such that
• fully residually R–groups belong to R and
• (split) extension of R–groups with finite kernel belong to R.
Let G be a (split) extension with a finitely generated residually finite kernel K and a
quotient group Q in R:
1→ K ↪→ G  Q→ 1.
Then G belongs to R.
Proof. Take a finite subset S ⊆ G. Define F := S−1S and choose a finite index subgroup
K0 E K such that F ∩K0 = {e}. Since K0 is of finite index and K is finitely generated one
can find a finite index subgroup K1 6 K0 which is characteristic, that is, Aut(K)–invariant.
Then, for N := K/K1 (note that, by construction, K1 is normal in G, and hence, in K), the
group G1 := G/K1 has the following properties:
(1) S injects into G1;
(2) G1 is a (split) extension of Q with finite kernel N .
Denote the quotient map by π : G  G1. Let s1, s2 ∈ S. If π(s1) = π(s2), then π(s−11 s2) = e.
However, s−11 s2 ∈ S
−1S ∩ ker(π) = F ∩ K1 ⊆ F ∩ K0 = {e}. Thus, s1 = s2 and we
obtain (1). For (2), we observe that the quotient G1/N is isomorphic to G/K  Q and
N is finite by construction. In addition, a splitting homomorphism G/K → G induces a
splitting homomorphism G1/N  (G/K1)/(K/K1) → G/K1.
By (2) and the hypothesis on R, G1 belongs to R. As a consequence, since S is an
arbitrary finite subset of G, the group G is fully residually R. Thus, G belongs to R. 
Lemma 2. Assume that R is a class of groups such that
• direct products of a finite and of an R–group belong to R and
• all finite index subgroups of R–groups belong to R.
Then any semidirect extension with finite kernel of an R–group belongs to R.
126
On approximation properties of semidirect products
Proof. Let G be a split extension with finite kernel of an R–group. Then G admits a
retraction ϕ : G → H on its finite index subgroup which belongs to R. Consider the
homomorphism G ↪→ Sym(G/H) × H : g 7→ ( f H 7→ g f H, ϕ(g)). It is injective and the
image has finite index in Sym(G/H) × H which belongs to R. Thus, G belongs to R. 
Let us apply our theorem to groups mentioned above. We begin with surjunctive
groups.
The concept of surjunctivity was introduced by Gottschalk [7] in topological dynamics
in 1973. Applied to actions of a discrete group G, it can be viewed as an analogue of
co–Hopf property of groups and of Artinian modules.
Definition 3. A continuous dynamical system (X,G) is called surjunctive if every
continuous injective map f : X → X commuting with the action of G is surjective.
Definition 4. Given a finite alphabet Σ the associated Bernoulli shift is the space ΣG
of Σ–valued functions on G.
Definition 5. A group G is called surjunctive if the Bernoulli shift ΣG is surjunctive for
any finite alphabet Σ.
All sofic groups are surjunctive [8, 14] and the class of sofic groups is the largest
known class of surjunctive groups. It is still unknown whether every group is surjunctive
and whether there exists a non-sofic surjunctive group. In particular, it is not clear whether
there exists a group G such that {0, 1}G is surjunctive but {0, 1, 2}G is not.
A very few affirmative results are known regarding the class of surjunctive groups:
• Finite groups are surjunctive [3, Proposition 3.1.3].
• Every subgroup of a surjunctive group is surjunctive [3, Proposition 3.2.1].
• Locally surjunctive groups are surjunctive [3, Proposition 3.2.2]. In particular,
S∞ =
⋃
n→∞ Sn, the group of finitely supported permutations of N, is surjunctive.
• Fully residually surjunctive groups are surjunctive [3, Lemma 3.3.4]. In particular,
Abelian groups and free groups are surjunctive.
Lemma6. Virtually surjunctive groups are surjunctive. In particular, semidirect extension
with finite kernel of a surjunctive group is surjunctive.
Proof. Assume that H 6 G is of finite index and H is surjunctive. Then we have
ΣG =
(
ΣG/H
)H , as H–spaces, and every G–invariant map is H–invariant. This yields the
claim. 
127
G. Arzhantseva & Ś. R. Gal
It is unknown whether a product of two surjunctive groups is surjunctive [14]. The
question is open both for direct and free products. A particular case of free products with
group of integers Z is open as well.
The concept of surjunctivity extends to a more broad context where the space ΣG
of Σ–valued functions on G is considered for Σ being an object in a surjunctive category C.
This naturally leads to the notion of C-surjunctive group [4]. For instance, the above
definition of surjunctive groups corresponds to C being the category of finite sets and if
C is the category of finite-dimensional vector spaces over an arbitrary field one obtains
the notion of linear surjunctivity [3, Chapter 8.14].
Without going into technical details (precise definitions are given in [4]) a surjunctive
category is a category where every injective C-endomorphism (self-map) is surjective. Let
us give examples of surjunctive categories (see [4, Example 7.3] and references therein):
• the category of finite sets,
• the category of finite-dimensional vector spaces over an arbitrary field,
• the category of left Artinian modules over an arbitrary ring, of finitely generated
left modules over an arbitrary left-Artinian ring,
• the category of affine algebraic sets over an arbitrary uncountable algebraically
closed field, or
• the category of compact topological manifolds.
Let us note that all the categories above are closed under taking finite direct products
(cf. property (FP) in [4, Section 3.1]).
Our Lemma 6 remains valid for the class of C-surjunctive groups with C any surjunctive
category closed under taking finite direct products; in particular, for any C from the list
above.
We extend Malcev’s result who considered the case of R being the class of residually
finite groups [10].
Theorem 7. Let R denote one of the following classes of groups:
• residually amenable groups,
• groups locally embeddable into finite groups (that is, LEF–groups),
• initially subamenable groups (in other words, LEA–groups),
• sofic groups,
• surjunctive groups, or
• C-surjunctive groups, where C is a surjunctive category closed under taking
finite direct products.
128
On approximation properties of semidirect products
Then any semidirect extension of a group from class R with residually finite, finitely
generated kernel belongs to R.
Proof. Fully residually surjunctive groups are surjunctive by [3, Lemma 3.3.4]. A
straightforward generalization of the proof of [3, Lemma 3.3.4] shows that fully residually
C-surjunctive groups are C-surjunctive. Analogous statements for other classes are
obvious.
To check if an arbitrary semidirect extension with finite kernel of an R–group belongs
to R, we apply Lemma 2 in case of residually amenable, LEF, or LEA groups, Lemma 6
in case of surjunctive groups and also, as mentioned above, in case of C-surjunctive
groups, as well as [3, Proposition 7.5.14] in case of sofic groups.
Thus, all the discussed classes satisfy the hypothesis of Lemma 2 and, therefore, that
of the split case of Theorem 1. 
Question 8. Let G be a non-split extension with a finite kernel and a surjunctive (resp.
C-surjunctive) quotient. Is G surjunctive (resp. C-surjunctive)?
A positive answer to this question, combined with the non-split case of Theorem 1,
will imply that all such extensions with residually finite, finitely generated kernel are
surjunctive (resp. C-surjunctive). An analogous question for residually amenable, LEF, or
LEA groups has a negative answer (since there exist extensions with a finite kernel and a
residually finite quotient which have Kazhdan’s property (T), and therefore are not LEA,
e.g. [6, 11]). It remains open for sofic and hyperlinear groups, cf. [1, Question 4.1]. In a
particular instance when the kernel is a finite cyclic group, a positive answer to Question 8
would imply the surjunctivity of Deligne’s non-residually finite central extensions [6], of
its SLn analogues [11] and of its recent arithmetic analogues [9].
Acknowledgments
The authors thank Benji Weiss for a helpful conversation and for pointing out that the
surjunctivity question was open even for direct products of a surjunctive group with Z.
References
[1] Goulnara Arzhantseva, Federico Berlai, Martin Finn-Sell, and Lev Glebsky. Unre-
stricted wreath products and sofic groups. Int. J. Algebra Comput., 29(2):343–355,
2019.
[2] Valerio Capraro and Martino Lupini. Introduction to sofic and hyperlinear groups
and Connes’ embedding conjecture, volume 2136 of Lecture Notes in Mathematics.
Springer, 2015. with an appendix by Vladimir Pestov.
129
G. Arzhantseva & Ś. R. Gal
[3] Tullio Ceccherini-Silberstein and Michel Coornaert. Cellular automata and groups.
Springer Monographs in Mathematics. Springer, 2010.
[4] Tullio Ceccherini-Silberstein and Michel Coornaert. Surjunctivity and Reversibility
of Cellular Automata over Concrete Categories. In Trends in Harmonic Analysis,
volume 3 of Springer INdAM Series, pages 91–133. Springer, 2013.
[5] Christophe Champetier and Vincent Guirardel. Limit groups as limits of free groups.
Isr. J. Math., 146:1–75, 2005.
[6] Pierre Deligne. Extensions centrales non résiduellement finies de groupes arithmé-
tiques. C. R. Math. Acad. Sci. Paris, 287(4):A203–A208, 1978.
[7] Walter Gottschalk. Some general dynamical notions. In Recent advances in topo-
logical dynamics, volume 318 of Lecture Notes in Mathematics, pages 120–125.
Springer, 1973.
[8] Mikhael Gromov. Endomorphisms of symbolic algebraic varieties. J. Eur. Math.
Soc., 1(2):109–197, 1999.
[9] Richard M. Hill. Non-residually finite extensions of arithmetic groups. Res. Number
Theory, 5(1): article ID 2 (27 pages), 2019.
[10] Anatoliı̆ Mal′cev. On homomorphisms onto finite groups. Ivanov. Gos. Ped. Inst.
Uč. Zap. Fiz.-Mat. Fak., 18:49–60, 1956.
[11] John J. Millson. Real vector bundles with discrete structure group. Topology,
18(1):83–89, 1979.
[12] Vladimir G. Pestov. Hyperlinear and sofic groups: a brief guide. Bull. Symb. Log.,
14(4):449–480, 2008.
[13] Anatoliı̆ M. Vershik and Evgeniı̆ I. Gordon. Groups that are locally embeddable in
the class of finite groups. Algebra Anal., 9(1):71–97, 1997.
[14] Benjamin Weiss. Sofic groups and dynamical systems. Sankhyā, Ser. A, 62(3):350–
359, 2000. Ergodic theory and harmonic analysis (Mumbai, 1999).
Goulnara Arzhantseva
Universität Wien
Fakultät für Mathematik
Oskar–Morgenstern–Platz 1,
1090 Wien, Austria
goulnara.arzhantseva@univie.ac.at
Światosław R. Gal
Uniwersytet Wrocławski
Instytut Matematyczny
pl. Grunwaldzki 2/4,
50–384 Wrocław, Poland
sgal@math.uni.wroc.pl
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