A note on finiteness properties of graphs of groups
By Frédéric Haglund and Daniel T. Wise
Abstract
We show that if $G$ is of type $\mathcal{F}_n$, and $G$ splits as a finite graph of groups, then the vertex groups are of type $\mathcal{F}_n$ if the edge groups are of type $\mathcal{F}_n$.
1. Introduction
The purpose of this note is to explain the following which is proven in Theorem 5.1:
For $n=1$, Theorem 1.2 is the following. It is obtained in Reference DD89 but the idea goes back to Stallings’ binding ties Reference Sta65, and the theorem is surely older.
Theorem 1.4 appears to be a “folk theorem”. Dunwoody suggested to us that it could be obtained by applying Reference DD89, Thm VI.4.4 followed by a folding sequence Reference BF91. There is a proof of it by Guirardel-Levitt who obtained a more powerful version relating to relative properties Reference GL17, Prop 4.9.
Theorem 1.2 is the converse to the following classical statement, which holds since a graph of $K(\pi ,1)$ spaces with $\pi _1$-injective attaching maps is a $K(\pi ,1)$. See Theorem 2.3.
2. Examples and a problem
There are many examples illustrating the failure of $\mathcal{F}_n$ for the vertex or edge groups of an $\mathcal{F}_n$ group that splits as a graph of groups. The most highly studied examples arise in the course of studying finiteness properties of the subgroup $N$ arising from a short exact sequence:
$$\begin{equation*} 1\rightarrow N \rightarrow G \rightarrow \mathbb{Z} \rightarrow 1 \end{equation*}$$
In this case, $G \cong N\rtimes \mathbb{Z}$ can be thought of as an HNN extension where the edge and vertex groups are copies of $N$.
There are many examples where $G$ is $\mathcal{F}_n$ but $N$ fails to be $\mathcal{F}_n$. Stallings and then Bieri Reference Sta63Reference Bie81 understood the motivating case where $G=(F_2)^n$ and the homomorphism sends the generators of each $F_2$ factor to the generators of $\mathbb{Z}$. Remarkably, while $G$ is $\mathcal{F}_n$, the subgroup $N$ is $\mathcal{F}_{n-1}$ but not $\mathcal{F}_n$. This led to the Morse theory of Bestina-Brady providing a plethora of similar examples Reference BB97.
In fact, in this context, it is difficult for $N$ to be $\mathcal{F}_n$ without $\mathsf{cd}(N)<\mathsf{cd}(G)$, as explained by Bieri Reference Bie81.
In parallel with Theorem 1.2 but generalizing from trees to CAT(0) cube complexes, we propose two formulations of a converse which we believe are equivalent:
The following shows that assuming all codimension-1 hyperplane stabilizers are $\mathcal{F}_n$ does not ensure that vertex stabilizers are $\mathcal{F}_n$.
3. Background
Choose a generator $\alpha$ of $\mathsf{H}_n(S^n)$. The Hurewicz homomorphism$h:\pi _n(X,x)\rightarrow \mathsf{H}_n(X)$ is defined by viewing any based $n$-sphere$f:(S^n,s)\rightarrow (X,x)$ as an $n$-cycle via $h(f)=[f_*(\alpha )]$.
Let $D^n\subset S^n$ be a hemisphere containing the basepoint $s$, and let $[\alpha ]$ represent a generator of $\mathsf{H}_n(S^n,D^n)$. The relative Hurewicz homomorphism$h:\pi _n(X,A,a)\rightarrow \mathsf{H}_n(X,A)$ is defined by viewing any relative based $n$-sphere$f:(S^n,D^n,s)\rightarrow (X,A,a)$ as an $n$-cycle via $h(f)=[f_*(\alpha )]$. We use the following relative form of the Hurewicz Theorem Reference Hat02, Thm 4.37 adapted to the simpler case where $A$ is simply-connected (to ensure injectivity of $h$).
For low dimensions we use that path connectivity is detected by $\widetilde{\mathsf{H}}_0=0$, as well as the following well-known statement Reference Hat02, Thm 2A.1:
4. Useless tree definitions and useful subtree lemmas
Our seemingly artificial requirement that $X\rightarrow T'$ is surjective on vertices allows us to naturally recover $T$ from $X$ as the nerve of the covering by vertex spaces.
Finally, as $\phi :X\rightarrow T'$ is $G$-equivariant and surjective on vertices we have $\operatorname {Stabilizer}(X_v)=G_v$ and $\operatorname {Stabilizer}(X_e)=G_e$ for each vertex $v$ and edge $e$ of $T$.
5. Main result
In this section we prove our main result expressed in terms of actions on trees instead of graphs of groups.
Before proceeding to the main part of the proof, we explain the final consequence:
Acknowledgment
We are grateful to Ross Geoghegan for a helpful comment.
Mladen Bestvina and Noel Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), no. 3, 445–470, DOI 10.1007/s002220050168. MR1465330, Show rawAMSref\bib{BestvinaBrady97}{article}{
label={BB97},
author={Bestvina, Mladen},
author={Brady, Noel},
title={Morse theory and finiteness properties of groups},
journal={Invent. Math.},
volume={129},
date={1997},
number={3},
pages={445--470},
issn={0020-9910},
review={\MR {1465330}},
doi={10.1007/s002220050168},
}
Reference [BF91]
Mladen Bestvina and Mark Feighn, Bounding the complexity of simplicial group actions on trees, Invent. Math. 103 (1991), no. 3, 449–469, DOI 10.1007/BF01239522. MR1091614, Show rawAMSref\bib{BestvinaFeighn91}{article}{
label={BF91},
author={Bestvina, Mladen},
author={Feighn, Mark},
title={Bounding the complexity of simplicial group actions on trees},
journal={Invent. Math.},
volume={103},
date={1991},
number={3},
pages={449--469},
issn={0020-9910},
review={\MR {1091614}},
doi={10.1007/BF01239522},
}
Reference [Bie81]
Robert Bieri, Homological dimension of discrete groups, 2nd ed., Queen Mary College Mathematical Notes, Queen Mary College, Department of Pure Mathematics, London, 1981. MR715779, Show rawAMSref\bib{BieriBook81}{book}{
label={Bie81},
author={Bieri, Robert},
title={Homological dimension of discrete groups},
series={Queen Mary College Mathematical Notes},
edition={2},
publisher={Queen Mary College, Department of Pure Mathematics, London},
date={1981},
pages={iv+198},
review={\MR {715779}},
}
Reference [BW13]
Hadi Bigdely and Daniel T. Wise, Quasiconvexity and relatively hyperbolic groups that split, Michigan Math. J. 62 (2013), no. 2, 387–406, DOI 10.1307/mmj/1370870378. MR3079269, Show rawAMSref\bib{BigdelyWiseAmalgams}{article}{
label={BW13},
author={Bigdely, Hadi},
author={Wise, Daniel T.},
title={Quasiconvexity and relatively hyperbolic groups that split},
journal={Michigan Math. J.},
volume={62},
date={2013},
number={2},
pages={387--406},
issn={0026-2285},
review={\MR {3079269}},
doi={10.1307/mmj/1370870378},
}
Reference [DD89]
Warren Dicks and M. J. Dunwoody, Groups acting on graphs, Cambridge Studies in Advanced Mathematics, vol. 17, Cambridge University Press, Cambridge, 1989. MR1001965, Show rawAMSref\bib{DicksDunwoody1989}{book}{
label={DD89},
author={Dicks, Warren},
author={Dunwoody, M. J.},
title={Groups acting on graphs},
series={Cambridge Studies in Advanced Mathematics},
volume={17},
publisher={Cambridge University Press, Cambridge},
date={1989},
pages={xvi+283},
isbn={0-521-23033-0},
review={\MR {1001965}},
}
Reference [Geo08]
Ross Geoghegan, Topological methods in group theory, Graduate Texts in Mathematics, vol. 243, Springer, New York, 2008, DOI 10.1007/978-0-387-74614-2. MR2365352, Show rawAMSref\bib{GeogheganBook2008}{book}{
label={Geo08},
author={Geoghegan, Ross},
title={Topological methods in group theory},
series={Graduate Texts in Mathematics},
volume={243},
publisher={Springer, New York},
date={2008},
pages={xiv+473},
isbn={978-0-387-74611-1},
review={\MR {2365352}},
doi={10.1007/978-0-387-74614-2},
}
Reference [GL17]
Vincent Guirardel and Gilbert Levitt, JSJ decompositions of groups(English, with English and French summaries), Astérisque 395 (2017), vii+165. MR3758992, Show rawAMSref\bib{GuirardelLevitt2017}{article}{
label={GL17},
author={Guirardel, Vincent},
author={Levitt, Gilbert},
title={JSJ decompositions of groups},
language={English, with English and French summaries},
journal={Ast\'{e}risque},
number={395},
date={2017},
pages={vii+165},
issn={0303-1179},
isbn={978-2-85629-870-1},
review={\MR {3758992}},
}
Reference [GM18]
Daniel Groves and Jason F. Manning, Hyperbolic groups acting improperly, 2018.
Reference [Hat02]
Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR1867354, Show rawAMSref\bib{Hatcher2002}{book}{
label={Hat02},
author={Hatcher, Allen},
title={Algebraic topology},
publisher={Cambridge University Press, Cambridge},
date={2002},
pages={xii+544},
isbn={0-521-79160-X},
isbn={0-521-79540-0},
review={\MR {1867354}},
}
Reference [HR17]
G. Christopher Hruska and Kim Ruane, Connectedness properties and splittings of groups with isolated flats, 2017.
Reference [Mol68]
D. I. Moldavanskiĭ, The intersection of finitely generated subgroups(Russian), Sibirsk. Mat. Ž. 9 (1968), 1422–1426. MR0237619, Show rawAMSref\bib{Moldavanski68}{article}{
label={Mol68},
author={Moldavanski\u {\i }, D. I.},
title={The intersection of finitely generated subgroups},
language={Russian},
journal={Sibirsk. Mat. \v {Z}.},
volume={9},
date={1968},
pages={1422--1426},
issn={0037-4474},
review={\MR {0237619}},
}
Reference [Sta63]
John Stallings, A finitely presented group whose 3-dimensional integral homology is not finitely generated, Amer. J. Math. 85 (1963), 541–543, DOI 10.2307/2373106. MR158917, Show rawAMSref\bib{Stallings63}{article}{
label={Sta63},
author={Stallings, John},
title={A finitely presented group whose 3-dimensional integral homology is not finitely generated},
journal={Amer. J. Math.},
volume={85},
date={1963},
pages={541--543},
issn={0002-9327},
review={\MR {158917}},
doi={10.2307/2373106},
}
Reference [Sta65]
John R. Stallings, A topological proof of Grushko’s theorem on free products, Math. Z. 90 (1965), 1–8, DOI 10.1007/BF01112046. MR188284, Show rawAMSref\bib{Stallings65}{article}{
label={Sta65},
author={Stallings, John R.},
title={A topological proof of Grushko's theorem on free products},
journal={Math. Z.},
volume={90},
date={1965},
pages={1--8},
issn={0025-5874},
review={\MR {188284}},
doi={10.1007/BF01112046},
}
Show rawAMSref\bib{4234060}{article}{
author={Haglund, Fr\'ed\'eric},
author={Wise, Daniel},
title={A note on finiteness properties of graphs of groups},
journal={Proc. Amer. Math. Soc. Ser. B},
volume={8},
number={11},
date={2021},
pages={121-128},
issn={2330-1511},
review={4234060},
doi={10.1090/bproc/81},
}
Settings
Change font size
Resize article panel
Enable equation enrichment
(Not available in this browser)
Note. To explore an equation, focus it (e.g., by clicking on it) and use the arrow keys to navigate its structure. Screenreader users should be advised that enabling speech synthesis will lead to duplicate aural rendering.