Graphs with polynomial growth are covering graphs

Graphs and Combinatorics (1992) 8:233-241

Graphs and Combinatorics �9 Springer-Verlag 1992

Graphs with Polynomial Growth Are Covering Graphs

C.D. Godsil and N. Seifter*

Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

Abstract. Let X be a connected locally finite transitive graph with polynomial growth. We show that there exist infinitely many finite graphs Y1, Y2 .... such that X is a covering graph of each of these graphs and every Yk, k > 2, is covering graph of the graphs YI . . . . . Yk-1. IfX is in addition s-transitive for some s > 2 the graphs Y~ can be assumed to be at least s-transitive.

1. Terminology and Introduction

By X(V,E) we denote a graph with vertex-set V(X) and edge-set E(X). Graphs considered in this paper are undirected, locally finite and contain neither loops nor multiple edges. By A U T(X) we denote the g roup of all au tomorphisms of X and id denotes the identity mapping. We say that a group G < AU T (X) acts transitively on X if there is a g ~ G for every v, w E V(X) such that g(v) = w. If such a group exists we call X transitive. A sequence (v0, vl . . . . . v~) of s + 1 vertices is called an s-arc if for each i, (vi_ 1, vi) is an edge of X and v , ~ vi+l. A graph X is s-transitive if A U T(X) acts transitively on the set of s-arcs in X. For convenience we sometimes regard a single vertex as a 0-arc and say that a transitive graph is 0-transitive.

If the stabilizer Gv <- G of a vertex v ~ V(X) consists of the identity only, then G acts semiregularly on X. If G in addit ion acts transitively on X then we say that it acts regularly on X.

Two one-way infinite paths P and Q are called equivalent in X, in symbols P ~x Q, if there is a third path R which meets bo th of them infinitely often (cf. [6], p. 127). The equivalence classes with respect to ~x are called ends. Obviously the au tomorphisms of X also act on the set of ends of X. A two-way infinite path is called a 2-path.

If G < A U T ( X ) acts transitively on X, then an imprimitivity system of G on X is a part i t ion z of V(X) into subsets called blocks, such that every element of G is a permutat ion of the blocks of ~. A m o n g imprimitivity systems we include the parti- t ion of V(X) into singletons and into V(X) itself. If z is a parti t ion into blocks of a group G which does not act transitively on X then �9 is called a block system of G

* Permanent address: Inst. f. Mathematik, Montanuniversitaet LeOben, A-8700 Leoben, Aus- tria. Work of both authors was in part supported by NSERC grant A5367.

234 C.D. Godsil and N. Seifter

on X. In both cases the quotient graph X~ is defined as follows: V(X 0 is the set of blocks and two vertices v~, w~ e V(XO are adjacent in X, if and only if (v, w) e E(X) for at least two vertices v ~ v , w ~ w~. By G~ we denote that group acting on X~ which is induced by G. Clearly it is a homomorphic image of G and G~ < AUT(X)~ < AUT(X~).

Let H be a subset of the group G, where 1 6 H and H = H -1. Then the Cayley graph C(G, H) of G with respect to H is defined on the vertex-set V(C(G, H)) = G and the edge-set

E(C(G,H)) = {(g, gh)lg ~ G,h ~ H}.

This graph is connected if H generates G and locally finite if H is finite. Furthermore we mention that G itself acts regularly on C(G, H) by left multiplication.

The growth function of a graph X, with respect to a vertex v ~ V(X) is defined by fx(v, 0) = 1 and

fx(v,n) = [{w e V(X)ld(v,w) < n}l, n E

where d(v, w) denotes the distance between v and w. If X is transitive the growth function clearly does not depend on a particular vertex v, therefore we denote it by fx(n). If there are constants c and d such that fx(n) <- cn a holds for all integers n > 1 then we say that X has polynomial growth.

These definitions coincide with those given for groups (see e. g. [5]). By the above definition of Cayley graphs it is also obvious that we can identify the growth functions of a group G and its Cayley graph with respect to some generating set H. If a finitely generated group G has polynomial growth, i.e. fG(n) < cn n, we know from [5] that G is almost nilpotent, which means that it contains a nilpotent normal subgroup of finite index. Furthermore this, together with a result due to H. Bass (see [1], Theorem 2), implies that there always exist constants el, Cz such that c I n d < f~(n) < c 2 n a for some integer d. This well defined integer is called the growth degree dG of G. We also mention that the growth degree does not depend on the generating se t, although the constants e 1 and e z depend on it. For more details we refer to [11].

A deep result of Trofimov (see [10], Theorem 2) implies that the same also holds for the growth functions of transitive graphs with polynomial growth. Hence we call the least integer d such that fx(n) <_ cn a holds for some constant c, the growth degree d x of X.

For the definition and basic properties of the fundamental group ~(X, v) of X at v e V(X) we refer to [7]. As in [3], p . denotes the induced group homomorphism rc(X~,v)~n(Xz,w) if p, p: X1 ~ X 2 , is a map such that p(V(X1))~_ V(X2) and p(v) = w.

If Vl, v2 ~ V(X) and 2 is a homotopy class of path s from va to v 2 then ~b~ denotes the group isomorphism r z(X, vl) ~ zc(X, v2) such that ~bx(/~) = 2-1p2.

Let G <_ AUT(X). A subgroup F _< rc(X, Vl) is G-admissible if for every g e G and every homotopy class 2 from v I to v2 = g(vl) we have g.(F) = ~bz(F). If G = AUT(X) we say just admissible instead of AUT(X)-admissible.

Let ~o, ~p: X 1 ~ X2, denote a homomorphism and let S(v), v e V(X~), be the star consisting of v and all edges incident to v. If q~(S(v)) is isomorphic t ~ S(v), for every v ~ V(X~), we call q~ a covering map and X1 a covering graph of X 2.

Infinite Covering Graphs 235

An orbit O of a subset H c_ A U T(X) is called a covering orbit if no edge of X joins two vertices of O. A pair O1, 02 of orbits of H on X is a covering pair if every vertex of O1 is adjacent to at most one vertex of 02 and vice versa.

In [9] it was shown that connected locally finite graphs with polynomial growth and valency at least three cannot be 8-transitive. Also a sketch of a construction of s-transitive graphs with polynomial growth as covering graphs of finite s-transitive graphs was given. Hence the question arises if there exist s-transitive graphs with polynomial growth which are not covering graphs of finite s-transitive graphs? In this paper we give a negative answer to this question, showing that for every connected locally finite graph X with polynomial growth there exist infinitely many finite graphs Y1, Y 2 , . . - such that X is covering graph of every Y~, i > 1. It is also shown that every Yk, k > 2, is covering graph of the graphs Y1, Y2 . . . . . Yk-l" If X is in addition s-transitive for some s > 2, then it is possible to show that the Y~ are also at least s-transitive.

For group-theoretic terminology and basic results we refer to [8].

2. Preliminary Results

The following characterisation of the automorphism groups of graphs with polynomial growth is crucial for the proofs given in this paper:

Theorem 2.1 (Trofimov [10]). Let X be a connected locally finite graph with polynomial growth and let G <_ AUT(X) act transitively on X. Then there exists an imprimitivity system z of G on X with finite blocks such that G~ is a finitely generated almost nilpotent group and the stabilizer of a vertex of X~ in G~ is finite.

We emphasize that Theorem 2.1 does not imply that the automorphism groups of graphs with polynomial growth always have polynomial growth (cf. [9]). In [9] we also proved the following result about s-transitivity of graphs with polynomial growth:

Theorem 2.2 (Seifter [9]). Let X be a connected locally finite s-transitive graph with polynomial growth and valency at least three. Then s <_ 7.

Examples of s-transitive graphs with polynomial growth and 0 _< s _< 7 can be constructed as covering graphs of finite s-transitive graphs. Since these examples give the motivation for the investigations carried out in this paper we describe them in detail here.

Proposition 2.3. For every s >_ 0 there exist connected locally finite s-transitive graphs with polynomial growth which are covering graphs of finite s-transitive graphs.

Proof. In [23 the following construction of a covering graph of a graph X with respect to a group G is given: Each edge (u, v) of X gives rise to two 1-arcs, (u, v) and (v, u). By S(X) we denote the set of 1-arcs and by a: S(X) ~ G we denote a mapping such that or(u, v) = (a(v, u)) -1 for all (u, v) e S(X). The covering graph J~ = )~(G, a)


of X with respect to G is defined on the vertex-set V(X) = G x V(X) and two vertices (gl, u), (g2, v) e V(3~) are joined by an edge if and only if (u, v) ~ S(X) and 92 = a l a(u, v).

I f X is a finite s-transitive graph with s > 8 we know from [,12] that X is a cycle. Since a 2-path is covering graph of every cycle our assertion obviously holds in this case.

Let X be a finite s-transitive graph where 0 < s < 7. It is well known (cf. [-7]) that the fundamental group re(X, u) of X at u is a free group on the edges of E(X)kE(T) (which are called chords) as generators, where T denotes a spanning tree of X. (We can also regard zc(X, u) as the group generated by the closed walks with base point u which are associated with the edges of E(X)kE(T). For details we again refer to [-7].)

We now consider the group A = zc(X, u)/[rc(X, u), z (X, u)]. It is a free finitely generated abelian group where the generators a I . . . . , a,, a~ -1 . . . . . a71 of A are the images of the generators gl . . . . . g,, g-;i . . . . . g;1 of re(X, u) under the homomorphism ~p from re(X, u) onto A. Hence A -~ JEt Let T be a spanning tree of X and let a, a: X ~ A, be a function which maps those elements of S(X) which correspond to edges of E(X) \E(T) onto the generators of A and the edges of T onto the unit element of A. It is easy to see that the covering graph X(A, a), which is constructed accordingly to the above rules, is a connected locally finite graph which has the same growth degree as A. It remains to show that )~(A, a) is s-transitive.

In I-3], Theorem 3, Djokovi6 proved that a covering graph ,Y of an s-transitive graph X is s-transitive if and only if the group F = p,(n(X,~)) is an admissible subgroup of zr(X, u), where p denotes a covering map and ~ ~ p-1 (u). In addition Theorem 4 of [-3] says that all characteristic subgroups of ~(X, u) are admissible. Hence it is sufficient to show that F is a characteristic subgroup of re(X, u).

To do this we regard the fundamental groups M X, u) and zt(X, ~) to be generated by the closed walks at u and ~, respectively. Let K = ((1, u), (gt, vt), . . . , (g,, v,) =

= . a~.~=l, e j~{- -1 ,1} , (1, u)) denote a closed walk in .g. Since A is abelian gn a~( .. ,, only holds if for every a~, the generator ai~ also occurs with the power - ek in g,. This implies that p(K) is a closed walk at u in X which is mapped onto the unit element of A by ~o. Hence p , (/~)s Ire(X, u), re(X, u)].

On the other hand every element h ~ l-re(X, u), re(X, u)] obviously gives rise to a closed walk in X(A, or). Hence F = [,r~(X, u), n(X, u)] holds, which completes the proof. []

3. Graphs with Polynomial Growth

Considering Proposition 2.3 it is a natural question to ask if there exist connected locally finite s-transitive graphs with polynomial growth which are not covering graphs of finite graphs? As the following results show, the answer to this question is negative. In particular if s > 2 it is not difficult to characterise those graphs as covering graphs of finite graphs.

Theorem 3.1. Let X be a connected locally finite s-transitive graph with polynomial growth and let s >>_ 2. Then there is an infinite sequence Y1, II2, Y3 . . . . of finite graphs


such that (1) X is covering graph of every Yi, i >_ 1. (2) Each Yi is at least s-transitive. (3) Each Yk, k >_ 2, is covering graph of the graphs Y1 . . . . . Yk-1.

Proof. If s >_ 8, we know from Theorem 2.2 that X has valency 2, which implies that it is a 2-path. In this case assertions (1) and (2) obviously hold since a 2-path is covering graph of every finite cycle. If the finite cycles Y~, i _> 1, in addition satisfy [ V(Yi+I)] = 2L V(Y0], then assertion (3) also holds.

Let 2 < s < 7. Then we know from [9], Proposition 4.5, that AUT(X) is almost nilpotent. We first assume that AUT(X) contains a nontrivial bounded automorphism of finite order. (An automorphism g ~ A U T(X) is called bounded if there is an integer k o such that d(v, g(O) < ko holds for all v ~ V(X).) Let Bo(X) denote the set of bounded automorphisms of finite order of X. From [4], Theorem 3, we know that B o(X) forms a normal subgroup of A U T(X) which acts with finite orbits on X. Hence the orbits of Bo(X) on X give rise to an imprimitivity system ~ of AUT(X) on X with finite blocks, Clearly no edge of X joins two vertices which are i~ the same block of z since the existence of such an edge would immediately imply that X is not 1-transitive. Now suppose there is a vertex v ~ V(X) which is adjacent to two vertices w o, w~ which are in the same block w,. Since X is a locally finite infinite graph it obviously contains a 2-arc P which meets three distinct blocks of ~. But then no automorphism of X maps the 2-arc (Wo, v, wt) onto the 2-arc P, a contradiction. Hence no vertex of X is adjacent to more than one vertex of the same block of T. Let (x, y) E E(X) where x and y are in different blocks of 3. Since these blocks are orbits of Bo(X ) every vertex x ' which is in the same block as x is adjacent to a vertex y ' which is in the same block as y. So the homomorphism q~o: X ~ X~ is a covering map. Also A U T(X)~ is almost nilpotent and acts s-transitively on X~ which can be shown using the methods of the proof of Theorem 4.3 in I-9] (Theorem 2.2 of this paper).

Let No now denote a normal nilpotent subgroup of finite index of A U T(X)~. As No is a finitely generated nilpotent group, the elements of finite order of No give rise to a finite normal subgroup of N o. Since N O acts with finitely many orbits on X~, it immediately follows that all elements of finite order of N O are bounded automorphisms of X~. But as N O <_ AUT(X)~ the unit element is the only element of N O which gives rise to a bounded automorphism of finite order of X,. Hence N O is a finitely generated torsion free nilpotent group. Now let G = AUT(X), and Y = X,. Let

No = L1 t > L 2 I> " " t> L, = {1},r > 2,

denote the lower central series of No. Since L 2 has growth degree less than that of No (cf. [1], Theorem 2) it acts with infinitely many orbits on Y. (If N O is abelian we have L 2 = {1}.) Furthermore L 2 <~ No, hence the orbits of L 2 on Y give rise to a block system e of No on Yand the group A = No~L2 acts with infinitely many infinite orbits on Y~. Since A is a finitely generated torsion free abelian group, A ~ 7/" for some m, 1 < m < dNo. Hence the group (q17/) m, for some integer q~ > 3, acts with at least three orbits on Y~. Then the group N* = (g ~ No[~(g ) ~ (q17/)"), where 0 is the homomorphism from N o onto A, also acts with at least, three orbits on E


Furthermore [No:N~'] is finite which implies that [G:No*] is also finite. So G contains a normal subgroup N1 of finite index (the intersection of all conjugates of N~' in G) which acts with finitely many but at least three orbits on Y. Hence the orbits of NI on Y give rise to an imprimitivity system e 1 of G on Y such that Y1 = Y~, is a finite graph. Using the methods of the proof of Theorem 4.3 in [9] it is again possible to show that G~, acts s-transitively on I11. For the same reason as before, no edge of Y can join two vertices which are in the same block of sl. Since N~ acts with at least three orbits on Y and Y is connected and at least 2-transitive it is again easy to see that no vertex of Y is adjacent to more than one vertex of the same block of e~. Therefore the orbits of N 1 on Y also give rise to a covering map ~0~ from Y onto Yr. So ~0~ o ~o o is a covering map from X onto Y~.

Repeating this for N~ instead of N O we obtain a subgroup N2 <~ N1, which also is a normal subgroup of finite index in G. Continuing this construction we at last get an infinite sequence of normal subgroups N~ t> N2 t> ... of G. Constructing the quotient graphs of Y with respect to the orbits of those normal subgroups, respectively, we obtain an infinite sequence I11, Y2 . . . . of finite graphs which satisfy assertions (1) and (2).

Let Nk now denote one of those normal subgroups of G. By the above construction we know that for every l, 1 < l < k - 1, the group Nt/N, is a normal subgroup of G/N k which acts with at least three orbits on Yk. Since G/Nk acts s-transitively, s _> 2, on Yk the above arguments again imply that Yk is a covering graph of every Yz-

[]

In the proof of Theorem 3.1 we have shown that the graphs Y1, Y2, . . . , are at least s-transitive. If they also have valency at least three and s = 7 holds, the result of R. Weiss [12], which says that such graphs cannot be 8-transitive, implies, that all those finite graphs are exactly 7-transitive. If s < 7 this need not hold. We even do not know if there always is at least one graph among the Y~, i > 1, which is not t-transitive for some t > s. Below we give an example of a 0-transitive graph with linear growth, which is a covering graph of finite 0-, 1- and 2-transitive graphs. This example indicates that it is a nontrivial problem to find assumptions such that those finite graphs are exactly s-transitive.

Let X be the Cayley graph C(G,H) of the group G = Z x Z 5 with respect to H = {a, a- l , c} where ( a ) ~ Z and ( c ) -~ Zs. By (a i, ci), j e Z, 0 < i < 4, we denote the vertices of C(G, H).

~ [a-1,1) , (a'l, c) f /a'1,c2; (a~,c3; (a.1,c4)

t :1,1) (1 ,c)

Fig. 1


Clearly every set Tj = {(a j, 1) . . . . . (a ~, c4)}, j ~ Z, separates the two ends of C(G, H) and the T~ are also minimal with respect to this property. If we now assume X to be t-transitive for some t _> 1 then there must be an automorphism g e AUT(X) which maps the edge ((1, 1), (1, c)) onto the edge ((1, 1), (a, 1)). But then the set 9(To) cannot separate the two ends of X, a contradiction. Hence X cannot be 1-transitive.

By {v o . . . . , v4} we now denote the vertex-set of the complete graph K 5. Define a map ~p: X ~ Ks, as follows: (a) ~p(aJ, c k) = v i i f j _> 0 and k - (i + 2j)(5). (b) ~p(aJ, c k) = v i i f j < 0 and k -- (i + lajl)(5). Obviously cp is a covering map and K 5 is 2-transitive.

The graphs C1 and C2 given by Figure 2 are Cayley graph s of Z 3 x Z 5 and Z 5 x Zs, respectively. Obviously C(G, H) also is a covering graph of these graphs. The edge (v o, v l )~ E(C~) is contained in the triangle (v o, v~, rE,/)0) but there is no triangle containing the edge (v o, v3). Hence Ca is not 1-transitive but of course it is 0-transitive.

c~

vo

V 3

Fig. 2

The graph C2 is obviously 1-transitive but as simple considerations show, the 2-arc (w o, wl, w2) cannot be mapped onto the 2-arc (Wo, Wl, w3). Hence it is not 2-transitive. In the sequel we show that s-transitive graphs with s _< 1 and polynomial growth can also be characterised as covering graphs of finite graphs. But as the situation is much more involved in this case it was not possible to prove exactly the same result as in the case s _> 2. Also the methods of the proof are quite different.

Theorem 3.2. Let X be a connected locally finite s-transitive graph with polynomial 9rowth and let 0 <_ s <_ 1. Then there exist infinitely many finite graphs Y1, Y2, ... such that (1) X is coverin9 9raph of every Yi, i >_ 1, and (2) Each Yk, k >_ 2 is coverin9 9raph of the 9raphs Ya, Y2 . . . . . Yk-1.

Proof. Let AUT(X) first be uncountable and let z be the imprimitivity system given by Theorem 2.1. It was shown in [9], Corollary 2.7, that we can always choose z such that its blocks coincide with the orbits of Bo(X ) on X.


If A U T ( X ) already is a finitely generated almost nilpotent group then z again denotes the imprimitivity system of A UT(X) on X which is induced by the orbits of Bo(X ) on X (which may be empty in this Case).

Although the homomorphism ~,: X ~ X~, need not be a covering map in the case s < 1, we first consider the graph X~. We also set Yo = X~ and G = AUT(X)~. Using the methods of the proof of [9], Theorem 4.3, it is again possible to show that G acts at least s-transitively on Yo. By Theorem 2.1 we also know that G is almost nilpotent and d a = dro. We first prove that our assertions hold for Yo.

Let No denote a nilpotent normal subgroup of finite index of G. It acts with finitely many orbits on u and as in the proof of Theorem 3.1 we know that it is torsion free. By 01 . . . . , O, we denote the orbits o f N o on Yo and g~ denotes a central element of No. Such an element always exist since No is nilpotent.

Each orbit O~, 1 < j < n, splits into orbits of gl. We denote a typical orbit of 91 by oj, and its elements by v{. (Here v{+l = gl(v]).) Suppose there are vertices v], . . . . . v{m, m >_ 1, with distance less than three from v~. Clearly m is finite, since Yo is locally finite. Set pj = max~ <~_<,, li~] + 1. Then 9f j acts with pj orbits on oj such that none of those orbits contains a pair of vertices of distance less than three. Since 9~J is also central in No all g ~ N o act on the orbits of 9{ 'j on Oj. Hence no orbit of g~'J on Oj contains a pair u, v of vertices with d(u, v) < 3. Since we can find such a number pj for every orbit of No on Yo, all orbits ofg~ 1 (where ql = 1-~7=1 Pj) have this property. Hence every orbit of g~l on X~ is a covering orbit and every pair of orbits of O q' on X~ is a covering pair. We can therefore assume that the orbits of 91 itself have this property.

As <gl> ~ Z is also a normal subgroup of No, the orbits of gl on Yo give rise a to block system el of No on Yo. Then, because of the properties of the action of 91 on Yo, the homomorphism q~l, q~l: Yo ~ (Yo),~, is a covering map. Furthermore, minor modifications of the proof of Proposition 2.5 of [10] show, that dYoi < dro - 1, where Y1 = (Yo)~. Also Na = (No)~ is a finitely generated nilpotent group which acts with finitely many orbits on Y~. Hence we can again apply the above arguments, with NI and Ya instead of No and Yo, respectively, thus obtaining a graph Yz with dr~ < dr1 - 1 and a covering map q)2:Y1 ~ Y2 etcetera. Since dr6 is finite, we at last obtain a finite graph W such that the map ~o: Yo ~ W, q~ = ~o~ o ... o r x < dro, is a covering map. The maps q~, . . . , ~0~ are induced by the orbits of automorphisms 9~ . . . . . 9~ of infinite order of Yo, Ya . . . . . Yx-1 which are central in No, N1 . . . . , N~-I, respectively. We mention that the groups N1 . . . . , N~_~ need not be torsion free. To conclude that they contain central elements of infinite order we only need the assumption that they are finitely generated nilpotent groups of infinite order.

We now consider the group H <_ G = A UT(X)~ which is generated by preimages of9~, 92, . . . , 9~ and their inverses. It acts with finitely many orbits on Yo and every orbit of H on Yo is a covering orbit as well as every pair of orbits is a covering pair. Since H has finite index in G we can find a normal subgroup H* of G which acts with finitely many orbits on Yo. Taking H* as the intersection of all conjugates of H in G it is also clear that H* .~ H. Hence the orbits of H* on Yo give rise to a covering map from Yo onto a finite graph. Since H* < N O it is also nilpotent. Hence, applying the methods of the proof of Theorem 3.1, we again obtain infinitely many finite graphs W1, W2 . . . . which satisfy the assertions of that Theorem.


Let HI = H*, H~, H~ . . . . be those normal subgroups of G whose orbits give rise to the covering maps f rom Yo onto the finite graphs W1, WE, W3 . . . . . respectively. By H1, H2 . . . . we denote their pre images under ~-1. Clearly they need not necessar- ily act semiregularly on X. But the groups HI, as subgroups of the torsion free group No, act semiregularly on Yo. Hence it is possible to find subsets S t _ Hi such that for every h ~ HI there is a unique s ~ St with ~(s) = h. Fu r the rmore we choose the elements of S~+1 such that S~+1 - St holds. Since the orbits of every St on X contain at most one vertex of each block of z, they are covering orbits. Also every pair of orbits of St on X is a covering pair. For, if they are not, then it immediately follows that the orbits of the HI on Yo are also not covering orbits or covering pairs. Hence X is covering graph of all quot ient graphs Y1, II2 . . . . of X with respect to the orbits of $1, S 2 . . . . on X. Since the I411, W2 . . . . are all finite the graphs Y1, Y2 . . . . are clearly also finite.

Assert ion (2) then is an immedia te consequence of our assumpt ion that S~+1 ___ St. []

Finally we want to ment ion that Theo rem 3.1 holds for 0- and 1-transi~ve graphs, provided we allow the finite graphs Y1, Yz, . . . to contain loops and multiiale edges.

References

1. Bass, H.: The degree of polynomial growth of finitely generated nilpotent groups. Proc. London Math. Soc. (3) 25, 603-614 (1972)

2. Biggs, N.: Algebraic graph theory. Cambridge Tracts in Mathematics 67. Cambridge Univer- sity Press 1974

3. Djokovi6, D.Z.: Automorphisms of graphs and coverings. J. Com. Theory (B) 16, 243-247 (1974)

4. Godsil, C., Imrich, W., Seifter, N., Watkins, M.E., W0ess, W.: A note on bounded automorphisms of infinite graphs. Graphs and Combinatorics 5, 333-338 (1989)

5. Gromov, M.: Groups of polynomial growth and expanding maps. Inst. Hautes Etudes Sci. Publ. Math. 53, 53-78 (1981)

6. Halin, R.: Ueber unendliche Wege in Graphen. Math. Ann. 157, 125-137 (1964) 7. Imrich, W.: Subgroup theorems and graphs. Comb. Math. 5, Lect. Notes Math. 622, 1 27

(1977) 8. Schwerdtfeger, H.: Introduction to Group Theory. Noordhoff International Publishing, Ley-

den 1976 9. Seifter, N.: Properties of graphs with polynomial growth. J. Com. Theory (B), (in print)

10. Trofimov, V.I.: Graphs with polynomial growth. USSR Sbornik 51 405-417 (1985) 11. Wagon, S.: The Banach-Tarski Paradox. Encyclopedia of Mathematics 24, Cambridge Univ.

Press, 1985 12. Weiss, R.: The nonexistence of 8-transitive graphs. Combinatorica 1 (3) 309-311 (1981)

Received: May 23, 1989 Revised: July 15, 1991

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Graphs with polynomial growth are covering graphs