GENERALIZED ECHELON SUBGROUPS BRAHIM ABDENBI Abstract. A subgroupHof a free groupFisinert if for any subgroup KF

2025-05-06 0 0 925.68KB 22 页 10玖币

侵权投诉

GENERALIZED ECHELON SUBGROUPS

BRAHIM ABDENBI

Abstract. A subgroup Hof a free group Fis inert if for any subgroup K ⊂ F,

we have rank (H ∩ K)≤rank (K). It is compressed if rank (H)≤rank (K) when-

ever H ≤ K. In this paper, we introduce highly inert graph immersions and

show that they represent inert subgroups. We use the compressibility of inert

subgroups to prove new properties on label distributions in their corresponding

graphs. Our main result is the generalization of Rosenmann’s echelon subgroups,

which he showed to be inert using endomorphisms of free groups. We show that

the collection of echelon subgroups is a proper sub-collection of generalized ech-

elon subgroups. Using some techniques from Mineyev-Dicks proof of the Hanna

Neumann Conjecture, we show inertness of generalized echelon subgroups, thus

providing a new proof for inertness of echelon subgroups.

1. Introduction

The notion of inert subgroups was ﬁrst introduced in 1996 by Dicks and Ventura

in [DV96]. A subgroup Hof a free group Fis inert if for any subgroup K ≤ F,

the rank of H ∩ K is bounded from above by the rank of K. Such subgroups arise

as ﬁxed subgroups of of injective endomorphisms F → F. Although this area

of research was largely motivated by the work of Dyer and Scott [DS75] in 1975,

interest in subgroups of free groups and their ranks dates back to earlier works by

many including Nielsen, Schreier, and Howson.

In 1926, Nielsen and Schreier proved that subgroups of free groups are free

[MKS76] [Joh80]. In the case where the subgroup is of ﬁnite index, they gave

an explicit formula for computing its rank, namely the Nielsen-Schreier formula. In

1954, Howson [How54] showed that the intersection of ﬁnitely generated subgroups

is ﬁnitely generated. In particular, he showed that if Hand Kare subgroups of

ﬁnite ranks, rank (H) and rank (K) respectively, then the rank of their intersection

is bounded above by 2 rank (H) rank (K)−rank (H)−rank (K) + 1. Soon afterward,

Hanna Neumann [Neu90] improved on Howson’s bound by showing that the rank of

the intersection is bounded from above by 2 (rank (H)−1) (rank (K)−1) + 1, and

further conjectured an even lower bound of (rank (H)−1) (rank (K)−1) + 1, which

came to be known as the Hanna Neumann Conjecture. This conjecture was solved

by Friedman [Fri15] in 2011 and independently by I. Mineyev [Min12] in the same

year.

Date: October 18, 2022.

2020 Mathematics Subject Classiﬁcation. 20E05, 20E07.

Key words and phrases. Free groups; subgroups intersection; echelon subgroups; generalized

echelon subgroups, inert subgroups; compressed subgroups; Hanna Neumann Conjecture.

Research supported by NSERC.

arXiv:2210.08651v1 [math.GR] 16 Oct 2022

GENERALIZED ECHELON SUBGROUPS 2

In 1996 Dicks and Ventura [DV96] introduced the notion of inert subgroups and

showed that ﬁxed subgroups of injective endomorphisms of free groups are inert. In

the same article they introduced the notion of compressed subgroups. A subgroup

H ≤ F is compressed if for any subgroup K ≤ F, if H ≤ K then rank (H)≤rank (K).

Inert subgroups are compressed. However, it remains an open question whether

compressed subgroups are inert or not. In [Ros13], Rosenmann introduced eche-

lon subgroups and showed that they are inert. These subgroups arise as images of

special endomorphisms called 1-generator endomorphisms. The main result of this

paper is the generalization of echelon subgroups.

Inert subgroups are mainly studied using endomorphisms of free groups, since they

ﬁrst arose in [DV96] as ﬁxed subgroups of injective endomorphisms of free groups.

Our approach, however, is mostly graph theoretic. In Section 2, we establish nota-

tions, and recall some classical deﬁnitions and theorems regarding graphs and free

groups. In Section 3, we introduce the notion of highly inert immersions which

will allow us to construct examples of inert subgroups. We also show some proper-

ties of compressed graphs pertaining to the distribution of labels in their edge sets.

The main result in this section is the proof that compressed graphs admit maximal

essential sets that map injectively into the bouquet of circles. In Section 4 we intro-

duce the class of generalized echelon subgroups, and show that it properly contains

Rosenmann’s echelon subgroups. We give a short overview of Mineyev-Dicks’ proof

of the Hanna Neumann conjecture, and use some of its results to show inertness of

generalized echelon subgroups, thus providing a new proof for inertness of echelon

subgroups.

2. Preleminaries

2.1. Graphs and Morphisms. Adirected graph Γ is a 1-dimensional CW -complex.

The sets of its vertices and edges, denoted by Γ0and Γ1, are the 0-cells and open

1-cells, respectively. There exist two incidence maps o, τ : Γ1→Γ0mapping each

edge e∈Γ1to its boundary vertices,o(e), τ (e) which we refer to as the origin

and terminus of e, respectively. The edge eis oriented from o(e) to τ(e). A mor-

phism of graphs φ: Γ1→Γ2is a continuous map that sends vertices to vertices

and edges to edges homeomorphically. If a base vertex vis chosen in Γ1, then

φ: (Γ1, v)→(Γ2, φ (v)) is a based morphism. A bouquet of ncircles is a graph B

with a single vertex and nedges. For simplicity, a based morphism into Bis denoted

by (Γ, v)→B. A labelling of a graph Γ is a morphism `: Γ →B. A morphism

φ: Γ1→Γ2is label preserving if the following diagram commutes

Γ1Γ2

`1`2

A morphism is an immersion if it is locally injective. Unless otherwise speciﬁed, all

graphs Γ are compact and equipped with a ﬁxed labelling `: Γ →B, where `is an

immersion.

GENERALIZED ECHELON SUBGROUPS 3

Given a non-negative integer m, we denote by Imthe graph homeomorphic to the

interval [0, m]⊂Rwhere I0

m= [0, m]∩Zand I1

m={(i, i + 1) |0≤i≤m−1}. A

path pof length mjoining two vertices vand wis a morphism p:Im→Γ such

that p(0) = vand p(m) = w. When v=w,pis a closed path. In particular, p

is a cycle if it is closed and injective on (0, m). A cycle of length 1 is a loop. A

concatenation of two paths p:Im→Γ and p0:Im0→Γ is a path γ:Im+m0→Γ

such that γ|[0,m]=pand γ|[m,m+m0]=p0. Note that this requires p(m) = p0(0).

Asubgraph is a subcomplex. A graph is connected if any two of its vertices can be

joined by a path. A (connected) component is a maximal connected subgraph with

respect to inclusion. A forest is a graph that contains no cycles and a connected

component of a forest is a tree.

The link of a vertex v∈Γ0, denoted by link (v), is the set of all length 1 paths

starting at v

link (v) = {p:I1→Γ|p(0) = v}

A loop at vcontributes two paths to link (v). Observe that an immersion φ: Γ1→Γ2

induces injective maps link (v)→link (φ(v)) for all v∈Γ0

The degree of a vertex v, denoted deg (v), is the cardinality of link (v). If |link (v)|<

∞for all v∈Γ0then Γ is locally ﬁnite. In this paper, we only consider graphs

whose vertices have uniformly bounded degrees, ie there exists D≥0 such that

|link (v)| ≤ Dfor all v∈Γ0. A ﬁnite graph is a core if all its vertices have degrees

greater than or equal to 2. Any graph that is not a forest deformation retracts to

a core subgraph. A vertex vis a branching vertex if deg (v)≥3. Given a set of

vertices S, we deﬁne S∗as

S∗={v∈S|deg (v)≥3}.

In particular, Γ∗is the set of branching vertices of Γ.

The Euler characteristic of a compact graph Γ is χ(Γ) = Γ0−Γ1. Its reduced

rank, denoted by ]

rank (Γ), is X

Γi⊂Γ

max {0,−χ(Γi)}, where Γiare the components

of Γ. The rank of a component Γiis rank (Γi)=1−χ(Γi). Observe that ﬁnite

connected graphs have the same rank as their core subgraphs.

We now give a few results whose proofs are omitted but can be found in [Sta83].

Lemma 2.1. Composition of immersions is an immersion.

Lemma 2.2. Given an immersion of graphs φ: Γ1→Γ2and a vertex v∈Γ0, the

induced homomorphism of fundamental groups

φ∗:π1(Γ1, v)→π1(Γ2, φ (v))

is injective.

2.2. Foldings. If `: Γ →Bis not an immersion, then we call a pair of edges

(e1, e2)∈Γ1×Γ1admissible if

(1) o(e1) = o(e2) or τ(e1) = τ(e2), and

(2) `(e1) = `(e2).

GENERALIZED ECHELON SUBGROUPS 4

Folding

Figure 1. Folding of two edges

Afolding of Γ is a map

f: Γ →Γ/(e1∼e2)

realized by identifying e1and e2. For example, if `(e1) = `(e2) = a, then a folding

is the procedure shown in Figure 1.

Lemma 2.3. If (e1, e2)∈Γ1×Γ1is an admissible pair, then the folding

f: Γ →Γ/(e1∼e2)

is π1-surjective. Therefore, ﬁnite compositions of foldings are also π1-surjective.

In particular, if Γ is connected, then rank (Γ) ≥rank (Γ/(e1∼e2)).

2.3. Graphs and Subgroups of Free Groups. Let F=π1B. Finitely gener-

ated subgroups of Fcan be represented by immersions of ﬁnite graphs into B. We

summarize the algorithmic construction of such graphs below and refer the reader

to [Sta83] [KM02] for more details.

Let H=hh1, . . . , hri≤Fbe a ﬁnitely generated subgroup. We start by ﬁrst taking

a disjoint union of rcircles such that for each 1 ≤i≤r, the circle Ciis subdivided

to form a closed labelled cycle which reads as the generator histarting from some

ﬁxed vertex vi. We construct the based graph

(H, v) =

i=1

v1∼v2∼. . . ∼vr

where all chosen vertices are identiﬁed to a single vertex v. If (H, v) immerses into

Bthen we are done, otherwise perform all possible foldings until (H, v)→Bis an

immersion. The resulting graph is the desired one. In particular, π1(H, v) = H.

Henceforth, we will denote groups with script letters and their corresponding graphs

with regular capital letters; for example the subgroup His represented by the graph

Remark 2.4. If the generators hiare cyclically reduced then all vertices of Hhave

degrees ≥2 except for possibly the base vertex.

2.4. Fiber Product of Immersions. Let φ:H→Γ and ψ:K→Γ be two

immersions of ﬁnite graphs. The pullback of these two maps

A K

HΓ

β ψ

GENERALIZED ECHELON SUBGROUPS 5

also called the ﬁber product is the graph

A=H⊗ΓK

deﬁned as follows

(1) A0=H0×K0

(2) A1=(e1, e2)∈H1×K1|φ(e1) = ψ(e2)

The immersions α:A→K, β :A→Hare projections.

We are mainly interested in the case where Γ = Band φ, ψ are immersions.

Theorem 2.5. Let

A K

H B

β`K

be a pullback diagram of graph immersions `Hand `K. Let v0= (u, v)∈A0be such

that α(v0) = vand β(v0) = u. Deﬁne F=`K◦α=`H◦β. Then

F∗π1A, v0=`K∗(π1(H, u)) ∩`H∗(π1(K, v))

In other words, the based component of the ﬁber product is precisely the graph

representing the intersection of the fundamental groups of Kand H. Note that the

three subgroups all lie inside F.

Remark 2.6. The cardinality of intersections of ﬁbers is at most 1:

|α−1(v)∩β−1(u)| ≤ 1 and |α−1(e2)∩β−1(e1)| ≤ 1

That is, if two vertices (edges) in Aproject to the same vertex (edge) in Kthen they

must project to distinct vertices (edges) in H. This is because both `K◦α:A→B

and `H◦β:A→Bare immersions.

Two immediate corollaries to this theorem are

Corollary 2.7. The intersection of ﬁnitely generated subgroups is itself ﬁnitely gen-

erated. .

This result was ﬁrst obtained by Howson in [How54].

Corollary 2.8. If subgroups H,K ≤ F are ﬁnitely generated then as wvaries over

F, the subgroups K ∩ w−1Hwbelong to only a ﬁnite number of conjugacy classes of

2.5. The Combinatorial Gauss-Bonnet Theorem for Graphs. Given a graph

Γ, one can deﬁne the curvature at a vertex v∈Γ0according to the following formula:

κ(v) = π(2 −deg (v))

When Γ is ﬁnite, the Combinatorial Gauss-Bonnet theorem relates the Euler char-

acteristic of Γ to the curvature in the following way

Theorem 2.9.

2πχ (Γ) = X

v∈Γ0

κ(v)

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：

GENERALIZEDECHELONSUBGROUPSBRAHIMABDENBIAbstract.AsubgroupHofafreegroupFisinertifforanysubgroupKF,wehaverank(H\K)rank(K).Itiscompressedifrank(H)rank(K)when-everHK.Inthispaper,weintroducehighlyinertgraphimmersionsandshowthattheyrepresentinertsubgroups.Weusethecompressibilityofinertsubgroupstoprov...

展开>> 收起<<

GENERALIZED ECHELON SUBGROUPS BRAHIM ABDENBI Abstract. A subgroupHof a free groupFisinert if for any subgroup KF.pdf

共22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！

GENERALIZED ECHELON SUBGROUPS BRAHIM ABDENBI Abstract. A subgroupHof a free groupFisinert if for any subgroup KF

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: