NoteThe firefighter problem on polynomial and intermediate growth groups
Introduction
Let be a graph and let be a sequence of integers; an initial fire starts at a finite set of vertices; at each time interval , at most vertices which are not on fire become protected, and then the fire spreads to all unprotected neighbours of vertices on fire; once a vertex is protected or is on fire, it remains so for all time intervals. The graph has the -containment property if every initial fire admits a strategy consisting of protecting at most vertices at the th time interval so that the set of vertices on fire is eventually constant. We say that the graph has the -containment property if there is a constant such that has the -containment property. Understanding the relation between and its containment functions is considered an asymptotic version of the firefighting game introduced by Hartnell in [5].
In [3] Dyer, Martinez-Pedroza and Thorne prove the following upper bound:
Theorem [3] Theorem 2.3 Let be a connected graph with polynomial growth of degree at most . Then satisfies the –containment property.
This is a generalization of a theorem by Develin and Hartke [2], which showed this holds for . They also prove the following more general upper bound:
Theorem [3] Theorem 2.8 Let be a locally finite connected graph, let be some vertex in and let be its rooted growth function. Let denote its discrete 2nd derivative . If is non-negative and non-decreasing then has the -containment property with .
Essentially the containment strategy behind the above theorems consists of constructing a large (far enough) sphere around the initial fire. The problem of finding good (possibly matching) lower bounds remained quite open, and only partial results are known.
A first observation is that for general graphs, growth does not capture the correct containment. In fact there are bounded degree graphs of exponential growth where 1 firefighter is enough to contain any initial function. One such graph is the Canopy tree (start with and to each vertex connect a rooted binary tree of depth ).
However, one can hope that when the graph is symmetric enough, or specifically on Cayley graphs, one would have a stronger relation between growth and containment.
Develin and Hartke [2] conjectured that for , the following converse holds:
Conjecture [2] Conjecture 9 Suppose that . Then there exists some outbreak on which cannot be contained by deploying firefighters at time .
Partial results were proved by Dyer, Martinez-Pedroza and Thorne in [3]. To that end they studied a spherical version of isoperimetry, looking at the number of neighbours of subsets of inside . They proved that if satisfies a “spherical isoperimetric inequality” in the sense that every satisfies , then does not satisfy -containment for any for which . This allows to deduce that does not satisfy -containment, but fails to fully resolve the conjecture. One should also note that many groups do not satisfy this spherical isoperimetric inequality (e.g. the lamplighter group .)
Note that by results of [3] containment is a quasi-isometry invariant, in the following sense: define if there exists s.t. and if and . Theorem 4.4. of [3] states that if is quasi-isometric to , and satisfies the -containment property, then satisfies the -containment property for some . It is well-known that for groups, growth and isoperimetric profiles are also invariant under quasi-isometries and, in particular, their equivalence classes do not depend on the choice of finite symmetric generating set.
We are now ready to state our main Theorem, which settles the Develin-Hartke conjecture. In fact, we show that the conjecture holds not only for but for any Cayley graph of polynomial growth.
Theorem 1 Let be a Cayley graph satisfying for some . Then does not satisfy -containment for any .
Our method uses isoperimetry to get lower bounds. By known connections between growth and isoperimetry, we are able to translate this into lower bounds depending on the growth for any growth rate. For exponential growth groups it is not optimal, and in fact does not even give an exponential lower bound. (Whereas by the results of [6], who uses very different methods, the critical threshold for exponential containment coincides with the growth rate, see also earlier results for elementary amenable groups in [7]). However it does allow us to get a superpolynomial lower bound on any group of intermediate growth, answering a well known open question (See e.g. [6] and Question in [3]).
Corollary 1 Let be the Cayley graph of some intermediate growth group. Then does not satisfy polynomial containment of any degree.
Currently, in all examples of groups of intermediate growth for which concrete lower bounds on the growth are known, these groups satisfy a stretched exponential lower bound of the form for some (in fact, the famous gap-conjecture asserts that all groups of intermediate growth satisfy such a lower bound for some fixed ). Given such a lower bound we can strengthen our result:
Theorem 2 Let be a Cayley graph satisfying for some . Fix any , and let be some function satisfying . Then does not satisfy -containment.
In [4], it is proved an isoperimetric inequality for Grigorchuk groups that improves upon the one attained directly form its growth. We can apply these bounds to get an improved lower bound on containment for Grigorchuk groups. (See [4] for exact definitions).
Theorem 3 Let be a Grigorchuk group where is not eventually constant, then does not satisfy -containment for any .
Question Is the above bound tight? That is, does Grigorchuk’s group satisfy -containment for some ? For some for arbitrary ?
Section snippets
Proofs
We begin with some notation: let be some finitely generated group and fix some finite symmetric generating set . We will identify with its Cayley graph w.r.t. . Let be the volume growth of the group , and let denote the isoperimetric profile of , that is (where denotes the outer boundary of ). A group is said to satisfy a -dimensional isoperimetric inequality if for some constant .
Let denote the set of burning vertices at
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors would like to thank all the suggestions of the anonymous referee. While performing this research, G.A. and R.B. were supported by the Israel Science Foundation grant #575/16 and by GIF grant #I-1363-304.6/2016. G.K. was supported by the Israel Science Foundation grant #1369/15 and by the Jesselson Foundation .
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