THE TITS ALTERNATIVE FOR TWO-DIMENSIONAL ARTIN GROUPS AND WISES POWER ALTERNATIVE ALEXANDRE MARTIN

2025-04-26 0 0 705.43KB 24 页 10玖币

侵权投诉

THE TITS ALTERNATIVE FOR TWO-DIMENSIONAL ARTIN

GROUPS AND WISE’S POWER ALTERNATIVE

ALEXANDRE MARTIN

Abstract. We show that two-dimensional Artin groups satisfy a strengthen-

ing of the Tits alternative: their subgroups either contain a non-abelian free

group or are virtually free abelian of rank at most 2.

When in addition the associated Coxeter group is hyperbolic, we answer in

the aﬃrmative a question of Wise on the subgroups generated by large powers

of two elements: given any two elements a, b of a two-dimensional Artin group

of hyperbolic type, there exists an integer n≥1such that anand bneither

commute or generate a non-abelian free subgroup.

Contents

1. Introduction 1

2. Preliminaries on Artin groups and their Deligne complex 4

2.1. Artin groups 4

2.2. The (modiﬁed) Deligne complex 5

2.3. Dihedral Artin groups and some associated graphs. 6

2.4. Standard trees 8

3. Free subgroups generated by elliptic elements 8

4. The Tits Alternative 16

5. Subgroups generated by two powers 18

5.1. The coned-oﬀ Deligne complex 18

5.2. The proof of Wise’s Power Alternative 18

References 23

1. Introduction

Motivation and statement of results. The Tits Alternative was ﬁrst intro-

duced by Tits in [Tit72], where he proved that a ﬁnitely generated subgroup of

a linear group over a ﬁeld either contains a free group or is virtually solvable.

This striking dichotomy has since been established for a large class of groups. A

meta-conjecture asserts that groups that are non-positively curved in a suitable

sense satisfy the Tits Alternative. While this has been proved for certain classes

of groups (CAT(0) cubical groups [SW05, CS11], hierarchically hyperbolic groups

[DHS17, DHS20], etc.) it remains open in general, and even for CAT(0) groups

despite some recent progress [OP21].

In this article, we will focus on Artin (or Artin-Tits) groups, a large class of

groups introduced by Tits in [Tit66]. These groups generalise braid groups and

have strong connections with Coxeter groups. While the geometry and structure

of Coxeter groups are now well-understood, the geometry of Artin groups remains

elusive in general. It is conjectured that these groups are CAT(0), although this

is only known in very few cases (see for instance [Hae22, Conjecture A] and the

discussion thereafter for a list of partial results). Nonetheless, certain classes of

arXiv:2210.06369v2 [math.GR] 30 Aug 2023

2 ALEXANDRE MARTIN

Artin groups have been shown to be non-positively curved in a more general sense,

see for instance [HO20, HO21].

In light of the above, it is natural to ask whether all Artin groups satisfy the Tits

Alternative. This is known already for certain classes of Artin groups, including

Artin groups of spherical type [CW02, Dig03], Artin groups of FC type [MP20],

and two-dimensional Artin groups of hyperbolic type [MP22]. The ﬁrst goal of this

paper is to generalise the latter result to all two-dimensional Artin groups:

Theorem A. Let AΓbe a two-dimensional Artin group. Then every subgroup of

AΓeither contains a non-abelian free group or is virtually free abelian of rank at

most 2.

When proving the Tits Alternative, one is led to construct free subgroups of

the group under study, which is challenging even for CAT(0) groups. However, the

situation sometimes becomes much more manageable when looking at the subgroups

generated by suitably large powers. In the case of a hyperbolic group for instance,

it is a standard result that given two inﬁnite order elements a, b with disjoint limit

sets in the Gromov boundary of the group, there exists a positive integer nsuch

that anand bngenerate a free group.

For groups that are non-positively curved in a more general sense (including

CAT(0) groups, biautomatic groups, etc.), Wise asked in [Bes04, Question 2.7]

whether a similar “taming” phenomenon occurs, namely: given any two elements

a, b, does there exists a positive integer nsuch that anand bneither generate

a free group or a free abelian group? We will say that a group satisfy Wise’s

Power Alternative when such a result holds. Leary–Minasyan’s recent example

of a CAT(0) that is not biautomatic [LM21] also provides the ﬁrst example of a

non-positively curved group that does not satisfy this alternative. It is thus natural

to ask the following:

Question 1.1. Which (non-positively curved) groups satisfy Wise’s Power Alter-

native?

There are already a few known groups that satisfy this alternative. For in-

stance, Baudisch showed [Bau81] that right-angled Artin groups satisfy an even

stronger statement: two elements in a right-angled Artin group either commute or

generate a free group. In particular, Wise’s Power Alternative is satisﬁed by all

groups that virtually embed in a right-angled Artin group, such as Coxeter groups

[HW10]. In another direction, Koberda proved that two arbitrary elements in a

non-exceptional mapping class group admit powers that are contained in a right-

angled Artin subgroup [Kob12]. In particular, Wise’s Power Alternative also holds

for non-exceptional mapping class groups.

Since we believe Artin groups to be non-positively curved in general, we ask the

following:

Question 1.2. Which Artin groups satisfy Wise’s Power Alternative?

The second main result of this article is the following:

Theorem B. Two-dimensional Artin groups of hyperbolic type satisfy Wise’s Power

Alternative.

Note that XXL-type Artin groups are known to be CAT(0) [Hae22], and that

extra-large type Artin groups are known to be systolic [HO21], so the above result

provides a large new class of non-positively curved groups that satisfy Wise’s Power

Alternative. In a forthcoming article with Mark Hagen, we will prove Wise’s Power

Alternative for some other classes of Artin groups (alongside other groups) via their

actions on trees [HM23].

TITS ALTERNATIVE FOR TWO-DIMENSIONAL ARTIN GROUPS 3

Strategy of the proofs. Our proof of the Tits Alternative is geometric and re-

lies on the action of a two-dimensional Artin group on its Deligne complex (see

Deﬁnition 2.4), a particularly well-behaved CAT(0) simplicial complex.

In [MP22], the authors could exploit the dynamics of the acylindrical action

of two-dimensional Artin groups of hyperbolic type on a variation of their Deligne

complex to construct many free subgroups. This approach is no longer possible for

general two-dimensional Artin groups, and the strategy of this paper is therefore

quite diﬀerent. Instead, given a subgroup Hof AΓ, we will treat separate cases

depending on whether the non-trivial elements of Hall act loxodromically on the

Deligne complex, all act elliptically, or whether Hcontains both loxodromic and

elliptic elements. In particular, we provide a classiﬁcation result for subgroups of

AΓ(Proposition 4.1). A key case to consider is the case where Hcontains two

elements that act as elliptic isometries of the Deligne complex with disjoint stable

ﬁxed-point sets, where the stable ﬁxed-point set of an isometry is deﬁned as the

union of the ﬁxed-point sets of all its non-trivial powers. We recall some general

questions here:

Question 1.3. Let a,bbe two elliptic isometries of a CAT(0) space Xthat have

disjoint stable ﬁxed-point sets.

(i) Does there exist an element of ⟨a, b⟩that acts loxodromically on X?

(ii) Does there exists a positive integer nsuch that ⟨an, bn⟩is a free group?

Such questions have been studied in the literature, in particular in relation with

the question of the existence of inﬁnite torsion subgroups of CAT(0) groups (ques-

tion (i), see for instance [NOP22, Conjecture 1.5]), and with the Tits Alternative

(question (ii)). A key intermediate result in our proofs of Theorems A and B is the

following:

Proposition C. Let AΓbe a two-dimensional Artin group, and let a,bbe two

elements of AΓthat act elliptically on the Deligne complex DΓwith disjoint ﬁxed-

point sets. Then there exists a positive integer nsuch that ⟨an, bn⟩is a non-abelian

free group.

Moreover, an element of ⟨an, bn⟩acts loxodromically on DΓif and only if it is

not conjugated to a power of anor bn.

Our strategy to prove Proposition C is to construct a tree embedded in the

Deligne complex on which ⟨an, bn⟩acts and such that there is a bijection between

edges of that tree and the reduced words on {a±n, b±n}. To that end, we consider

a geodesic γbetween the ﬁxed-point sets of aand b, and we show that for some

large enough n, the geodesic segment γmakes an angle at least πwith any of its

⟨an⟩-translates or ⟨bn⟩-translates. The CAT(0) geometry of the Deligne complex

then guarantees that the resulting subspace obtained as the union of all ⟨an, bn⟩-

translates of γis a convex subtree of DΓ. (see Figure 1)

Controlling these angles requires a ﬁne understanding of the local structure of

the Deligne complex, and this problem can be translated into a problem about

dihedral Artin groups, a class that is very well understood.

In order to prove Theorem B, the additional assumption that AΓis of hyperbolic

type allows us to exploit the action of the Artin group on its coned-oﬀ Deligne com-

plex (see Deﬁnition 5.1), a variation of the Deligne complex introduced in [MP22]

on which AΓacts acylindrically. The additional dynamical properties of this action

are a key tool in proving that certain subgroups generated by large powers are free.

4 ALEXANDRE MARTIN

anγ

a2nγ

a2nbnγ

a2nb2nγ

Figure 1. A small portion representing the convex subtree of DΓ

obtained by gluing together all the ⟨an, bn⟩-translates of the ge-

odesic segment γbetween the ﬁxed-point sets Fix(a)(in yellow)

and Fix(b)(in blue).

Structure of the paper. In Section 2, we recall standard deﬁnitions and results

about Artin groups and their Deligne complexes. In Section 3, we prove the key

Proposition C on free subgroups generated by powers of elliptic elements with

disjoint ﬁxed-point sets. In Section 4, we prove Theorem A by ﬁrst providing a

classiﬁcation of the subgroups of a two-dimensional Artin group (Proposition 4.1).

Finally, in Section 5, we use the action of a two-dimensional Artin group of hyper-

bolic on its coned-oﬀ Deligne complex to prove Theorem B.

Aknowledgements. I would like to thank Piotr Przytycki, as this article grew

out of previous work and discussions with him. I would also like to thank Thomas

Delzant for providing a very short proof (and a much simpler one than the one

originally there) of Lemma 5.12. I also thank the anonymous referee for their

comments and careful reading of this paper.

This work was partially supported by the EPSRC New Investigator Award

EP/S010963/1.

2. Preliminaries on Artin groups and their Deligne complex

This preliminary section contains deﬁnitions of the main groups and complexes

used in this article, and recalls various results from the literature.

2.1. Artin groups.

Deﬁnition 2.1. Apresentation graph is a ﬁnite simplicial graph Γsuch that

every edge between vertices a, b ∈V(Γ) comes with a label mab ≥2. The Artin

group (or Artin-Tits group) associated to the presentation graph Γis the group

AΓgiven by the following presentation:

AΓ:=⟨a∈V(Γ) |aba · · ·

| {z }

mab

=bab · · ·

| {z }

mab

whenever a, b are connected by an edge of Γ⟩.

An Artin group on two generators a, b with mab <∞is called a dihedral Artin

group. Given an Artin group AΓ, the associated Coxeter group WΓis obtained

by adding the relations a2= 1 for every a∈V(Γ) . An Artin group is said to be of

hyperbolic type if the associated Coxeter group is hyperbolic, and of spherical

type if the associated Coxeter group is ﬁnite.

Deﬁnition 2.2. Given an induced subgraph Γ′⊂Γ, the subgroup of AΓgenerated

by the vertices of Γ′is called a standard parabolic subgroup. Such a subgroup is

isomorphic to the Artin group AΓ′by [vdL83], and conjugates of standard parabolic

subgroups are called parabolic subgroups.

TITS ALTERNATIVE FOR TWO-DIMENSIONAL ARTIN GROUPS 5

Deﬁnition 2.3. An Artin group is said to be two-dimensional if for every in-

duced triangle Γ′of Γ, the corresponding standard parabolic subgroup AΓ′is not

of spherical type.

2.2. The (modiﬁed) Deligne complex. We recall here some important complex

on which an Artin group acts.

Deﬁnition 2.4 ([CD95]).The (modiﬁed) Deligne complex of a an Artin group

AΓis the simplicial complex deﬁned as follows:

•Vertices correspond to left cosets of standard parabolic subgroups of spher-

ical type.

•For every g∈AΓand for every chain of induced subgraphs Γ0⊊· · · ⊊Γk

such that AΓ0, . . . , AΓkare of spherical type, we put a k-simplex between

the vertices gAΓ0, . . . , gAΓk.

In other words, the (modiﬁed) Deligne complex DΓis the geometric realisation of

the poset of left cosets of standard parabolic subgroups of spherical type.

The group AΓacts on DΓby left multiplication on left cosets.

Let us recall a few elementary properties about this action. Note that since

vertices of a simplex of DΓcorrespond to left cosets of pairwise distinct subgroups,

AΓacts on DΓwithout inversion: if an element of AΓglobally stabilises a simplex

of DΓ, it ﬁxes it pointwise. As a consequence, the stabiliser of a simplex of DΓ

corresponding to the chain gAΓ0⊊· · · ⊊gAΓkis the parabolic subgroup gAΓ0g−1.

In the rest of this article, we will simply speak of the Deligne complex instead

of the modiﬁed Deligne complex for simplicity.

For two-dimensional Artin groups, there is a simple description of the vertices

of the Deligne complex DΓ:

Lemma 2.5. Let AΓbe a two-dimensional Artin group. Then vertices of DΓare

of the following type:

•left cosets of the form gAΓ′=g{1}where Γ′is the empty subgraph. Such

vertices are called of type 0 and their stabiliser is trivial.

•left cosets of the form gAΓ′=g⟨a⟩where Γ′={a}is a single vertex of Γ.

Such vertices are called of type 1 and their stabiliser is inﬁnite cyclic.

•left cosets of the form gAΓ′=g⟨a, b⟩where Γ′is an edge between two

vertices a, b of Γ. Such vertices are called of type 2 and their stabiliser is

isomorphic to a dihedral Artin group.

In particular, the Deligne complex DΓis a simplicial complex of dimension 2. More-

over, it follows from the discussion above that the stabilisers of higher dimensional

simplices of DΓare as follows:

•The stabiliser of an edge of DΓis either inﬁnite cyclic or trivial.

•The stabiliser of a triangle of DΓis trivial.

While the geometry of Deligne complexes is still mysterious in general (and

strongly related to the K(π, 1)-conjecture for Artin groups, see [CD95]), the geom-

etry of Deligne complexes for two-dimensional Artin groups is well-understood:

Theorem 2.6 ([CD95]).Let AΓbe a two-dimensional Artin group. Its Deligne

complex can be endowed with a CAT(0) metric as follows: For a chain

g{1}⊊g⟨a⟩⊊g⟨a, b⟩(for g∈AΓand a, b adjacent vertices of Γ)

we identify the corresponding triangle of DΓwith a triangle of the Euclidean plane

E2with angles π

2mab ,π

2, and π

2−π

2mab at the vertices g⟨a, b⟩, g⟨a⟩, g{1}respectively,

and such that the edge between g⟨a⟩and g{1}has length 1.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

THETITSALTERNATIVEFORTWO-DIMENSIONALARTINGROUPSANDWISE’SPOWERALTERNATIVEALEXANDREMARTINAbstract.Weshowthattwo-dimensionalArtingroupssatisfyastrengthen-ingoftheTitsalternative:theirsubgroupseithercontainanon-abelianfreegrouporarevirtuallyfreeabelianofrankatmost2.WheninadditiontheassociatedCoxetergrou...

展开>> 收起<<

THE TITS ALTERNATIVE FOR TWO-DIMENSIONAL ARTIN GROUPS AND WISES POWER ALTERNATIVE ALEXANDRE MARTIN.pdf

共24页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

THE TITS ALTERNATIVE FOR TWO-DIMENSIONAL ARTIN GROUPS AND WISES POWER ALTERNATIVE ALEXANDRE MARTIN

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: