Random Group Actions on CAT0 Square Complexes Zachary Munro Abstract

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Random Group Actions on CAT(0) Square Complexes

Zachary Munro

Abstract

Generalizing ideas in [Jah11], we introduce the notion of progression

in CAT(0) square complexes. Using progression, we are able to build

on the proof strategy of [DGP11] to show any action of a random

group with seven or more generators on a CAT(0) square complex has

a global ﬁxed point.

1 Introduction

In this article, we begin the inquiry into the cubical dimension of random groups in the

Gromov density model, ﬁrst introduced in [Gro91]. Letting S={s1,··· , sn}be a set of

generators, a random group at density d∈(0,1) and length Lis given by the presentation

hS|Ri, where Ris a set of b(2n−1)dLc-many words chosen uniformly at random from the

set of reduced words of length `with alphabet S. One says that a random group has some

property Qwith overwhelming probability (w.o.p.) if the probability that hS|Risatisﬁes

Qapproaches 1 as L→ ∞. A number of results in the theory of random groups describe

“phase transitions” for particular properties Qwith respect to the density d. It is proven

in [Oll05], [Gro91] that for d > 1/2a random group is either trivial or Z2w.o.p., and for

d < 1/2a random group is inﬁnite, hyperbolic, and torsion-free w.o.p. Thus most of the

interesting theory occurs in the range d∈(0,1/2).

There have been various results regarding phase transitions associated with actions of a

random group on a CAT(0) cube complex. In the positive direction, in which one produces

actions on CAT(0) cube complexes, Ollivier and Wise [OW11]showed that random groups

with density d < 1

5act without global ﬁxed point on CAT(0) cube complexes of ﬁnite

dimension w.o.p. This result has been strengthened by MacKay and Przytycki [MP15], who

established the result for d < 5

24 , and Montee [Mon20], who improved the bound to d < 3

14 .

Recently, Ashcroft [Ash22] has proven that random groups at densities d < 1/4act without

global ﬁxed point on CAT(0) cube complexes of ﬁnite dimension w.o.p. In the negative

direction, it was shown by Żuk [Ż03] and Kotowski-Kotowski [KK13] that for d > 1/3a

random group has Kazhdan’s property (T) w.o.p. Groups with property (T) act always

with global ﬁxed point on CAT(0) cube complexes, following from an observation by Niblo

and Reeves [NR97]. The following conjecture was communicated to the author by Piotr

Przytycki 1.

1Note that 1

5is 4

5·1

4,5

24 is 5

6·1

4, and 3

14 is 6

7·1

arXiv:2210.06378v1 [math.GR] 12 Oct 2022

Conjecture. Random groups act without global ﬁxed point on ﬁnite dimensional CAT(0)

cube complexes at densities d < 1

4and have property (T) at densities d > 1

4w.o.p.

Note that Ashcroft’s work in [Ash22] completes the ﬁrst half of the conjecture.

For the purposes of this article, we concern ourselves with the dimension of those cube

complexes on which random groups act without global ﬁxed point. In [DGP11], it was shown

that random groups at any density act always with global ﬁxed point on 1-dimensional

CAT(0) cube complexes. We extend this result by proving the following.

Main Theorem. Random groups G=hS|Riwith |S| ≥ 7act always with global ﬁxed

point on 2-dimensional CAT(0) cube complexes w.o.p.

A natural way of proving that a random group hS|Ridoes not exhibit a particular

property Qis to show there exists a large collection of words Lin the alphabet Swhich

must be nontrivial if hS|Riwere to have property P. Because Lis large and Rconsists of

randomly sampled words, the probability that the intersection L∩Ris nonempty approaches

1 as L→ ∞. And thus we conclude that w.o.p. hS|Ridoes not have property P. This

approach was applied fruitfully in each of [DGP11], [Orl17], and [Jah11] to prove random

groups do not have particular properties. We are especially inspired by [Jah11], where it is

proved random groups act always with global ﬁxed point on R-trees.

In Section 2 we establish our notation regarding CAT(0) cube complexes and graphs. In

Section 3 we deﬁne checkpoint automata, a central tool in the paper. We prove a lemma

ensuring that the accepted language of a checkpoint automaton is large. Given a group action

G→Aut(X)on some set X, an element g∈Gacts nontrivially if its image in Aut(X)is

nontrivial. The lemmas proven in Section 4 are used to ensure that many elements of a

group action are nontrivial. It is here that we prove the central technical theorem. We are

immediately able to deduce a weakened form of our main theorem (Corollary 3.6), where

the number of generators depends on the density d. Section 5 is dedicated to removing the

dependency on d, using a modiﬁed version of a construction found in [DGP11]. Given a

random group G=hS|Riat some density d, one ﬁnds a ﬁnite index subgroup H < G

which is a quotient of a random group at density dwith exponentially more generators than

G. Property FC2is inherited by quotients by Lemma 2.1 and FC2is a commensurability

invariant by Lemma 2.2, and thus we deduce Ghas property FC2.

Acknowledgements. I would like to express my immense gratitude to my advisor Piotr

Przytycki, without whom none of this work would be possible. Thank you for posing this

problem to me. Thank you for our regular meetings. And thank you for encouraging me

when the math seemed hopeless. Thank you Adrien Abgrall for revising early drafts of this

paper.

2 Cube complexes and Graphs

We introduce our notation concerning CAT(0) square complexes and graphs below. For a

more thorough introduction to CAT(0) cube complexes, see [Sag14].

2.1 Actions on cube complexes

In our article, e

Xwill denote a CAT(0) square complex, and Swill be a set of formal letters

for which there exists a homomorphism FS→Aut( e

X), where FSis the free group on S.

For a hyperplane Hof e

X, we let N(H)denote the carrier of H. We let d(·,·)denote the

combinatorial metric on e

X1. For 0-cells x, y ∈e

X0, we let [x, y]denote a combinatorial

geodesic from xto y. Note that (e

X1, d)is not uniquely geodesic. Thus, if we ever assert

that [x, y]has some property, then it is implicit that the property holds independent of the

choice of geodesic. The intersection of a hyperplane with e

X1is a set of midpoints of 1-cells

in e

X1. The complement e

X−Hof a hyperplane has two components, called halfspaces. A

hyperplane Hseparates subsets A, B ⊂e

Xif Aand Bare contained in distinct components

of e

X−H. If His a hyperplane and x∈e

X0, we let d(x, H):= inf{d(x, h)|h∈H}

denote the smallest distance from xto H. For x∈e

X0recall that d(x, H)>1if and only if

x6∈ N(H)if and only if xis separated from Hby some other hyperplane. Also, for x, y ∈e

the distance d(x, y)is equal to the number of hyperplanes which separate x,y. Two distinct

hyperplanes H6=Wcross, denoted by HtW, when H∩W6=∅. If we write H∩W6=∅,

we leave open the possibility that H=W. Two distinct hyperplanes H6=Ware parallel if

H∩W=∅. This is denoted HkW.

An isometry sof a CAT(0) cube complex is a hyperplane inversion if it stabilizes some

hyperplane Hand interchanges its halfspaces. In this case, we say sinverts H. Any group

action is without hyperplane inversions after a subdivision of the cube complex.

We will make use of 2-dimensionality in a couple of ways. For one, no more than two

hyperplanes can pairwise intersect. Secondly, if HtW, then N(H)∩N(W) = Cwhere C

is a single square of e

X. In particular, for any 0-cell x∈N(H)−Cthere is a hyperplane

separating xand W.

A group Ghas property FCnif every action of Gby cubical isometries on an n-dimensional

CAT(0) cube complex has a global ﬁxed point. Ghas property FC∞if Ghas FCnfor every

n∈N. Note that FCn+1 ⊂FCnand that FC1is equivalent to Serre’s property FA. Similar

to property FA, the property FCnpasses to quotients since group actions can always be

pulled back along group homomorphisms. Consequently, we have the following.

Lemma 2.1. If Gis a quotient of some b

Gwith property FCn, then Ghas property FCn.

The commensurability invariance of property FA also generalizes to FCn. A ﬁnite set of

points in a CAT(0) cube complex has a unique barycentre, i.e. a point which minimizes the

sum of distances to all points in the set. The existence of barycentres immediately leads to

the following lemma.

Lemma 2.2. If a ﬁnite index subgroup H < G has property FCn, then Ghas property FCn.

Note the converse of the above lemma does not hold in general.

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摘要：

RandomGroupActionsonCAT(0)SquareComplexesZacharyMunroAbstractGeneralizingideasin[Jah11],weintroducethenotionofprogressioninCAT(0)squarecomplexes.Usingprogression,weareabletobuildontheproofstrategyof[DGP11]toshowanyactionofarandomgroupwithsevenormoregeneratorsonaCAT(0)squarecomplexhasaglobalxedpoint...

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Random Group Actions on CAT0 Square Complexes Zachary Munro Abstract

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