Martínez-Pedroza, Eduardo

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Institut Fourier — Université de Grenoble IActes du séminaire deThéorie spectraleet géométrieEduardo MARTÍNEZ-PEDROZAA note on the relation between Hartnell’s firefighter problem and growth ofgroupsVolume 33 (2015-2016), p. 55-60.<http://tsg.cedram.org/item?id=TSG_2015-2016__33__55_0>© Institut Fourier, 2015-2016, tous droits réservés.L’accès aux articles du Séminaire de théorie spectrale et géométrie(http://tsg.cedram.org/), implique l’accord avec les conditionsgénérales d’utilisation (http://tsg.cedram.org/legal/).cedramArticle mis en ligne dans le cadre duCentre de diffusion des revues académiques de mathématiqueshttp://www.cedram.org/Séminaire de théorie spectrale et géométrieGrenobleVolume 33 (2015-2016) 55-60A NOTE ON THE RELATION BETWEEN HARTNELL’SFIREFIGHTER PROBLEM AND GROWTH OFGROUPSEduardo Martínez-PedrozaAbstract. — The firefighter game problem on locally finite connected graphswas introduced by Bert Hartnell [6]. The game on a graph G can be described asfollows: let fn be a sequence of positive integers; an initial fire starts at a finite setof vertices; at each (integer) time n > 1, fn vertices which are not on fire becomeprotected, and then the fire spreads to all unprotected neighbors of vertices on fire;once a vertex is protected or is on fire, it remains so for all time intervals. Thegraph G has the fn-containment property if every initial fire admits an strategythat protects fn vertices at time n so that the set of vertices on fire is eventu-ally constant. If the graph G has the containment property for a sequence of theform fn = Cnd, then the graph is said to have polynomial containment. In [5],it is shown that any locally finite graph with polynomial growth has polynomialcontainment; and it is remarked that the converse does not hold. That article alsoraised the question of whether the equivalence of polynomial growth and polyno-mial containment holds for Cayley graphs of finitely generated groups. In this shortnote, we remark how the equivalence holds for elementary amenable groups andfor non-amenable groups from results in the literature.Let G be a connected and locally finite graph, and let d be a non-negative integer. The graph G satisfies a polynomial containment of degreeat most d if there exists a constant C > 0 such that any finite subset X0 ofvertices of G admits a {Cnd}-containment strategy, i.e., there is a sequence{Wk : k > 1} of subsets of vertices of G such that(1) for every n > 1, the set Wn has cardinality at most Cnd,(2) the sets Xn and Wn+1 are disjoint for n > 0, where Xn for n > 0 isdefined as the set of vertices at distance less than or equal to oneof Xn−1 and do not belong to W1 ∪ · · · ∪Wn.(3) there is an integer T > 0 such that Xn = XT for every n > T .This definition can be interpreted as the existence of winning strategiesfor a single player game: a fire starts at an arbitrary finite set of verticesX0, at each time interval the fire spreads to all unprotected vertices that2010 Mathematics Subject Classification: 20F65, 43A07, 57M07, 05C57, 05C10, 05C25.56 E. MARTÍNEZ-PEDROZAneighbor vertices on fire, and Xn is the set of vertices on fire at time n;the player aims to contain the fire by protecting at time n a set Wn of atmost Cnd vertices that are not on fire; once a vertex is protected or is onfire, it remains so for all time intervals. The graph G satisfies polynomialcontainment of degree at most d if there is C > 0 such that for any initialfire X0 the player has an {Cnd}-strategy to contain the fire.The notion of edge-path in a graph G defines a metric on the set of ver-tices by declaring distG(x, y) to be the length of the shortest path from xto y. The metric dist on the set of vertices of G is called the combinatorialmetric on G. The notion of quasi-isometry is an equivalence relation be-tween metric spaces which plays a significant role in the study of discretegroups, for a definition and overview see [3].Theorem 1 ([5, Theorem 8]). — In the class of connected graphs withbounded degree, the property of having polynomial containment of degreeat most d is preserved by quasi-isometry.For a connected and locally finite graph G with a chosen vertex g0, letβ(n) denote the number of vertices at distance at most n from g0. Recallthat the function β : N→ N is called the growth function of G with respectto g0. A graph G has polynomial growth of degree at most d if there is aconstant C > 0 such that β(n) 6 Cnd for each integer n. It is an exerciseto show that having polynomial growth of degree 6 d is independent of thebase vertex g0. Moreover, in the class of connected graphs with boundeddegree, it is a quasi-isometry invariant [3].The following relation between polynomial growth and polynomial con-tainment can be used to provide examples of graphs with polynomial con-tainment.Theorem 2 ([5, Theorem 3]). — Let G be a connected graph withpolynomial growth of degree d. Then G has polynomial containment ofdegree at most max{0, d− 2}.Remark 3. — The converse of Theorem 2 does not hold, there are con-nected graphs of bounded degree with subexponential growth and constantcontainment, see [5, Example 2.7]. The examples in the cited paper aregraphs of bounded degree with infinitely many vertices that separate thegraph into a finite subgraph and an infinite subgraph.In contrast, examples of locally finite infinite graphs without polyno-mial containment can be exhibited using the following proposition. By aδ-regular tree, we mean an infinite tree such that every vertex has degreeexactly δ.SÉMINAIRE DE THÉORIE SPECTRALE ET GÉOMÉTRIE (GRENOBLE)HARTNELL’S FIREFIGHTER PROBLEM AND GROWTH OF GROUPS 57Proposition 4 ([5, Corollary 11]). — If a graph G contains a subgraphquasi-isometric to the infinite δ-regular tree with δ > 3, then G does notsatisfy a polynomial containment property.For any finitely generated group G, any two Cayley graphs with respectto finite generating sets are quasi-isometric. In view of Theorem 1, we saythat G has polynomial containment of degree d if the Cayley graph of Gwith respect to a finite generating set has polynomial containment of degreed; and we say that G has polynomial containment if it has polynomialcontainment of degree d for some d. The question of whether the converseof Theorem 2 holds for finitely generated groups was raised in [5].Question 5 ([5, Question 12]). — In the class of finitely generatedgroups, is having polynomial growth of degree d equivalent to having poly-nomial containment of degree max{0, d− 2}?This note discusses the following weaker version of the above question.Question 6. — In the class of finitely generated groups, is having poly-nomial growth equivalent to have polynomial containment?In this note we observe that the answer is positive for elementaryamenable groups, and for non-amenable groups. Indeed, in the class offinitely generated groups, polynomial containment implies amenability; andin the class of finitely generated elementary amenable groups, polynomialcontainment implies polynomial growth. Below we explain this statements.Recall that the Cheeger constant h(G) of a locally finite graph G isdefined ash(G) = infK|∂K||K|where K is any non-empty finite subset of vertices of G, and ∂K is the setof edges of G with one endpoint in K and the other endpoint not in K.A locally finite graph G is amenable if h(G) = 0, and otherwise is callednon-amenable. In the class of graphs with bounded degree, being amenableis preserved by quasi-isometry; hence for a finitely generated group eitherCayley graphs with respect to finite generating sets are amenable, or allare non-amenable. This yields the notions of amenable and non-amenablein the class of finitely generated groups.Theorem 7 ([2, Theorem 1.1]). — Any non-amenable locally finitegraph contains a tree with positive Cheeger constant.Corollary 8. — Finitely generated groups with polynomial contain-ment are amenable.VOLUME 33 (2015-2016)58 E. MARTÍNEZ-PEDROZAProof. — Let Γ be a non-amenable finitely generated group, and G beits Cayley graph with respect to a finite generating set. By Theorem 7,the graph G contains a subgraph T which is a tree with positive Cheegerconstant. Observe that T is necessarily infinite; since T has bounded degreeand h(T ) > 0, it follows that there is an upper bound on the length ofembedded paths in T whose interior vertices have degree 2 in G. It followsthat T , and hence G, contains a subgraph quasi-isometric to the 3-regulartree. Therefore Proposition 4 implies that G does not have polynomialcontainment. The class of elementary amenable groups is the smallest class of groupscontaining all finite groups and all abelian groups, and closed under takingsubgroups, quotients, extensions, and direct unions.Theorem 9 ([4, Theorem 3.2′]). — A finitely generated elementary-amenable group is either virtually nilpotent or contains a free semigroupin two generators.Corollary 10. — Finitely generated elementary amenable groups withpolynomial containment are virtually nilpotent and hence they have poly-nomial growth.Proof. — Let Γ be a finitely generated elementary amenable group, andsuppose that it is not virtually nilpotent. By Theorem 9, there are elementsa, b ∈ Γ generating a free semigroup S in two generators. Consider theCayley graph G of Γ with respect to a finite generating set containing aand b. Then the elements of S span a subgraph T of G which is an infiniterooted tree, the root is the identity element of G, and any other vertex hasdegree 3. Observe that T is quasi-isometric to the 3-regular tree, and henceProposition 4 implies that G does not have polynomial containment.The result follows by invoking Wolf’s result that finitely generated vir-tually nilpotent groups have polynomial growth [10]. From Corollaries 8 and 10, it follows that polynomial growth is equivalentto polynomial containment in the class of finitely generated groups thatare elementary amenable or non-amenable. The class of non-elementaryamenable groups has been a subject of intense study since it was shownin the 1980’s that the class is non-empty with the appearance of Grig-orchuk’s group of intermediate growth [7]. The class of non-elementaryamenable groups contains groups of intermediate growth as wells as groupsof exponential growth. An example of a non-elementary amenable groupof exponential growth is the Basilica group; the group was introduced byGrigorchuk and Zuk [8] where they proved that it has exponential growthSÉMINAIRE DE THÉORIE SPECTRALE ET GÉOMÉTRIE (GRENOBLE)HARTNELL’S FIREFIGHTER PROBLEM AND GROWTH OF GROUPS 59and it does not belong to the class of elementary amenable groups; theproof that the group is amenable was found by Bartholdi and Virag [1].It is conceivable that Proposition 4 can be used to show that groups ofexponential growth have no polynomial containment, in other words, thatthe following folk question has a positive answer. A small variation of thisquestion was asked by A. Thom [9].Question 11. — Let G be a finitely generated non-elementary amen-able group of exponential growth, and let S be a finite generating set.Does the Cayley graph of G with respect to S contains a tree with positiveCheeger constant?Since the Cayley graph of a group of intermediate growth cannot containa tree with positive Cheeger constant, answering Question 6 is a more subtlequestion.Remark 12. — Laurent Bartholdi remarked that weakly branchedgroups have superpolynomial containment, and he conjectured that theirminimal firefighting sequence has grow as exp(√n). A weakly branchedgroup G contains a subgroup H that contains H × H as a proper sub-group; hence H is an infinite subgroup and the Cayley graph of G containsa d-dimensional grid up to quasi-isometry. Therefore G must have contain-ment of at least polynimial degree d− 2 for all d.Acknowledgements. The author acknowledges funding by the Natu-ral Sciences and Engineering Research Council of Canada (NSERC), andthanks Gidi Amir, Laurent Bartholdi, Florian Lehner, Pierre Will, and thereferee for comments and suggestions.BIBLIOGRAPHY[1] L. Bartholdi & B. Virág, “Amenability via random walks”, Duke Math. J. 130(2005), no. 1, p. 39-56.[2] I. Benjamini & O. Schramm, “Every graph with a positive Cheeger constant con-tains a tree with a positive Cheeger constant”, Geom. Funct. Anal. 7 (1997), no. 3,p. 403-419.[3] M. R. Bridson & A. Haefliger, Metric spaces of non-positive curvature,Grundlehren der Mathematischen Wissenschaften, vol. 319, Springer, 1999,xxii+643 pages.[4] C. Chou, “Elementary amenable groups”, Ill. J. Math. 24 (1980), no. 3, p. 396-407.[5] D. Dyer, E. Martínez-Pedroza & B. Thorne, “The coarse geometry of Hartnell?sfirefighter problem on infinite graphs”, Discrete Math. 340 (2017), no. 5, p. 935-950.[6] S. Finbow, B. L. Hartnell, Q. Li & K. Schmeisser, “On minimizing the effects offire or a virus on a network”, J. Comb. Math. Comb. Comput. 33 (2000), p. 311-322.VOLUME 33 (2015-2016)60 E. MARTÍNEZ-PEDROZA[7] R. I. Grigorchuk, “On the Milnor problem of group growth”, Dokl. Akad. NaukSSSR 271 (1983), no. 1, p. 30-33.[8] R. I. Grigorchuk & A. Żuk, “On a torsion-free weakly branch group defined bya three state automaton”, Int. J. Algebra Comput. 12 (2002), no. 1-2, p. 223-246.[9] A. Thom, “Trees in groups of exponential growth”, MathOverflow question, http://mathoverflow.net/q/60011 (version: 2013-05-09).[10] J. A. Wolf, “Growth of finitely generated solvable groups and curvature of Rie-mannian manifolds”, J. Differ. Geom. 2 (1968), p. 421-446.Eduardo Martínez-PedrozaMemorial UniversitySt. John’sNewfoundland A1C 5S7 (Canada)emartinezped@mun.caSÉMINAIRE DE THÉORIE SPECTRALE ET GÉOMÉTRIE (GRENOBLE)

A note on the relation between Hartnell’s firefighter problem and growth of groups

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A note on the relation between Hartnell’s firefighter problem and growth of groups

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