Growth rate for endomorphisms of finitely generated nilpotent groups and solvable groups

We prove that the growth rate of an endomorphism of a finitely generated nilpotent group equals to the growth rate of induced endomorphism on its abelinization, generalizing the corresponding result for an automorphism in [14]. We also study growth rates of endomorphisms for specific solvable groups, lattices of Sol, providing a counterexample to a known result in [5] and proving that the growth rate is an algebraic number.


Introduction
In the present paper, we study purely algebraic notions of growth rate and entropy for an endomorphism of a finitely generated group.
Let be a finitely generated group with a system S D ¹s 1 ; : : : ; s n º of generators. Let W ! be an endomorphism. For any 2 , let L. ; S/ be the length of the shortest word in the letters S [ S 1 which represents . Then the growth rate of is defined [2] to be GR. /´sup ® lim sup k!1 L. k . /; S / 1=k j 2 ¯: For each k > 0, we put and the algebraic entropy of is by definition h alg . /´log GR. /. The growth rate and hence the algebraic entropy of are well-defined, i.e., independent of the choice of a set of generators [7, p. 114]. It is immediate from the definition that the growth rate and the algebraic entropy for an endomorphism of a group are invariants of conjugacy of group endomorphisms. Furthermore, for any inner automorphism 0 by 0 , we have GR. 0 / D GR. / and h alg . 0 / D h alg . / ([7, Proposition 3.1.10]).
Consider a continuous map f on a compact connected manifold M , and consider a homomorphism induced by f of the group of covering transformations on the universal cover of M . Then the topological entropy h top .f / is defined. We refer to [7] for background. Several authors, among them R. Bowen in [2] and A. Katok in [8], have proved that the topological entropy h top .f / of f is at least as large as the algebraic entropy h alg . / D h alg .f / of or f .
The problem of determining the growth rate of a group endomorphism, initiated by R. Bowen in [2], is now an area of active research (see detailed description in [5,10] and references therein). For known properties of the growth of automorphisms of free groups, we refer to [1,12,13].
The purpose of this paper is first to study the growth rate of an endomorphism on a finitely generated nilpotent group. In [10,Theorem 1.2], it was proven that the growth rate of an automorphism of a finitely generated nilpotent group is equal to the growth rate of the induced automorphism on its abelianization. Our main result is a generalization of this result of [10] from automorphisms to endomorphisms, using completely different arguments. In Section 2, we recall some known results about the growth rate of a group endomorphism, sometimes correcting them. In Section 3, we refine the calculation in [2] of the growth rate for an endomorphism of a finitely generated torsion-free nilpotent group and prove that the growth rate is an algebraic integer.
Let be a finitely generated torsion-free nilpotent group, and let G be its Malcev completion. Let be an endomorphism of . Then extends uniquely to a Lie group homomorphism D of G, called the Malcev completion of . We call its differential D the linearization of . The main results are the following. Theorem 3.3. Let W ! be an endomorphism on a finitely generated torsionfree nilpotent group . Let G be the Malcev completion of . Then the linearization D W G ! G of can be expressed as a lower triangular block matrix with diagonal blocks ¹D j º so that In particular, GR. / is an algebraic integer.

Preliminaries
We shall assume in this article that all groups are finitely generated unless otherwise specified. For a given endomorphism W ! , if 0 is a -invariant subgroup of , we denote by 0 D j 0 the restriction of to 0 . If, in addition, 0 is a normal subgroup, we denote by O the endomorphism on = 0 induced by . Then the following are known; see for example [2,5].
Let W Z n ! Z n be an endomorphism yielding an integer matrix D. Then we have GR. / D sp.D/, the maximum of the absolute values of the eigenvalues of D.
Let S 0 be a finite set of generators for 0 , and let O S be a finite set of generators for the quotient group = 0 . Then it is possible to extend S 0 to a finite set S of generators for so that S is projected onto O S under the projection ! = 0 . For any 2 0 , it is true that L. ; S 0 / L. ; S/.
Consider the concentric balls B.n/ D ¹ 2 j L. ; S/ Ä nº for all n > 0, and the distortion function of 0 in which is defined as 0 .n/´max¹L. ; S 0 / j 2 0 \ B.n/º: The notion of distortion of a subgroup was first introduced by M. Gromov in [6]. We refer to [3] for our discussion. For two functions f; gW N ! N, we say that f 4 g if there exists c > 0 such that such that f .n/ Ä cg.cn/ for all n > 0. We say that two functions are equivalent, written f g, if f 4 g and g 4 f . The subgroup 0 of is undistorted if 0 .n/ n. The following facts about distortion can be found in [3].
If 0 is infinite, then it is true that n 4 0 .n/.
If OE W 0 < 1, then 0 is undistorted in . Assume 0 .n/ 4 n. By definition, there exists c > 0 such that 0 .n/ Ä c 2 n for all n > 0. For any 2 0 , let n D L. ; S/. Then L. ; S 0 / Ä 0 .n/ Ä c 2 n D c 2 L. ; S /: Thus L. ; S/ Ä c 2 L. ; S/ for all 2 0 . This inequality implies that, for all k > 0, and so GR. 0 / Ä GR. /. Consequently, we have the following lemma. Proof. Since 0 is undistorted in , we have from the definition that 0 .n/ 4 n. Now the proof follows from the above observation. we have GR. / D 1. Similarly, we have GR. j Z / D 0 and GR. j Z 2 / D 1. Notice further that Z 2 is a distorted subgroup of because Z 2 .n/ D 1 for all n.
Lemma 2.5. Let be an endomorphism of .
(2) If, in addition, 0 is a normal subgroup of , then Proof. If the -invariant subgroup 0 of is finite, then we can show easily that GR. 0 / is either 0 or 1 by taking a system of generators S 0 D 0 for 0 . We will show that GR. 0 / Ä GR. /. We may assume that GR. 0 / D 1. This implies that there is an element x 2 0 such that 0 n .x/ ¤ 1 for all n > 0. Considering any system of generators for which contains x, we can see right away that Remark 2.6. However, the above lemma is not true when 0 is infinite; see Example 2.7. Note further that if GR. / < GR. 0 /, then 0 is infinite.
The following is a well-known example about subgroup distortion.
Example 2.7. Let be the Baumslag-Solitar group Then S D ¹a; bº is a generating set for . Let 0 D hbi, and let S 0 D ¹bº. We observe that the subgroup 0 of is distorted. In fact, since b n k D a k ba k for all k > 0, we have that L.b n k ; S 0 / D n k and L.b n k ; S / D 2k C 1. If is an endomorphism of given by .b/ D b n and .a/ D a, then we can see that GR. 0 / D n and GR. / D 1.   This implies that GR. 0 / Ä GR. / Ä max¹GR. 0 /; 1º. Since 0 is not eventually trivial, Lemma 2.3 implies that GR. 0 / 1, and hence GR. / D GR. 0 /.
Next we assume that p 1. For each j D 1; : : : ; u, we write .ı j / D ı j 1 w 1 for some j 1 and w 1  When GR. / D 0, Lemma 2.3 says that is an eventually trivial endomorphism. Next we consider the case when GR. / D 1. From the definition, we can choose N > 0 so that, for m N , we have 1=2 m < L m . ; S /, which implies that L m . ; S/ 1 because L m . ; S/ is an integer. Therefore, for each m N , we can choose 2 S such that m . / ¤ 1. This shows that is not eventually trivial even though 0 is eventually trivial.
Before leaving this section, we observe the following elementary fact. we obtain a contradiction: GR. / D inf k K L k . ; S / 1=k < GR. /.

Finitely generated nilpotent groups
Consider the lower central series of a finitely generated group , where j D OE ; j 1 is the j -fold commutator subgroup j . / of . The endomorphism W ! induces endomorphisms Then it is known from [2] that GR. / GR. N j / 1=j for all j 1. The group is called nilpotent if j D 1 for some j . When c ¤ 1 but cC1 D 1, we say that it is c-step. ¹GR. N j / 1=j º: Recall, for example from [10, Proposition 3.1], that a finitely generated nilpotent group is virtually torsion-free. Thus there exists a finite-index, torsionfree, normal subgroup of . Following the proof of [11, Lemma 3.1], we can see that there exists a fully invariant subgroup ƒ of which is of finite index. Therefore, any endomorphism W ! restricts to an endomorphism 0 W ƒ ! ƒ. By Theorem 2.8, we may consider only the case when 0 is not eventually trivial, and hence we may assume that GR. / D GR. 0 /. Consequently, for the computation of GR. /, we may assume that is a finitely generated torsion-free nilpotent group.
Consider the lower central series of a finitely generated torsion-free c-step nilpotent group , For each j D 1; : : : ; c, we consider the isolator of j in , p j D p j´¹ x 2 j x k 2 j for some k 1º: Then it is known that p j is a characteristic subgroup of with OE p j W j finite. Furthermore, p j = j is precisely the set of all torsion elements in the nilpotent group = j , and p j = p j C1 Š Z k j for some integer k j > 0. Hence we obtain the adapted central series The following lemma plays a crucial role in our study of growth rates for endomorphisms of finitely generated nilpotent groups. Then there are finite sets T j D ¹ j1 ; : : : ; j k j º j such that (1) if p j W j ! j = j C1 denotes the projection, then p j .T j / is an independent set of generators for the finitely generated abelian group j = j C1 , (2) if j > 1, then every jr is of the form OE 1i ; j 1;` , Let G be the Malcev completion of a finitely generated torsion-free nilpotent group , and let be an endomorphism of . Then extends uniquely to a Lie group homomorphism D of G, called the Malcev completion of . We call its differential D the linearization of . Theorem 3.3. Let W ! be an endomorphism on a finitely generated torsionfree nilpotent group . Let G be the Malcev completion of . Then the linearization D W G ! G of can be expressed as a lower triangular block matrix with diagonal blocks ¹D j º so that In particular, GR. / is an algebraic integer.
Proof. Let be a finitely generated torsion-free c-step nilpotent group with the adapted central series denote the projection. We choose ¹T 1 ; : : : ; T c º as in Lemma 3.2. Since 2 is a fully invariant, finite-index subgroup of p 2 , it induces a short exact sequence Since p 2 = 2 is finite, it follows that can be regarded as the free part of the finitely generated abelian group 1 = 2 . Hence we can choose S 1 T 1 such that p 1 .S 1 / is an independent set of free generators of p 1 = p 2 and p 1 .T 1 S 1 / is an independent set of torsion generators of 1 = 2 . Next we consider the short exact sequence Since 2 = 3 p 2 = 3 , we obtain the following commutative diagram between exact sequences: where all vertical maps are inclusions of finite index. So we can choose S 2 T 2 such that p 2 .S 2 / is an independent set of free generators of the free abelian group . 2 p 3 /= p 3 and p 2 .T 2 S 2 / is an independent set of torsion generators of 2 = 3 . Note that S 2 2 p 2 . Because the right-most vertical inclusion is of finite index, we can choose S 2 p 2 such that q 2 .S 2 / is an independent set of free generators of p 2 = p 3 , and for each 2 2 S 2 , there are unique`2 1 and unique 2 2 S 2 such that 2`2 D 2 modulo p 3 . We remark also that #S 2 D #S 2 .
Continuing in this way, we obtain ¹S 1 ; : : : ; S c º ¹T 1 ; : : : ; T c º such that p j .S j / is an independent set of free generators of j = j C1 , p j .T j S j / is an independent set of torsion generators of j = j C1 , q j .S j / is an independent set of free generators of p j = p j C1 , for each j 2 S j p j , there exist unique`j 1 and j 2 S j such that The adapted central series of allows us to choose a preferred basis a of ; we can choose a to be ¹S 1 ; : : : ; S c º so that it generates and can be embedded as a lattice of a connected, simply connected nilpotent Lie group G, the Malcev completion of . Its Lie algebra G has a linear basis log a D ¹log S 1 ; : : : ; log S c º. From j`j D j mod p j C1 , we havè j log. j / D log. j`j / D log. j / mod j C1 .G/: This implies that ¹log S 1 ; : : : ; log S c º is also a linear basis of G. Let W ! be an endomorphism. Then induces endomorphisms : Moreover, any endomorphism on extends uniquely to a Lie group endomorphism D on G, the Malcev completion of . With respect to the preferred basis log a of the Lie algebra G of G, we can express the linearization D of as a lower triangular block matrix; each diagonal block D j is an integer matrix representing the endomorphism N For details, we refer to [9] for example. When the new basis ¹log S 1 ; : : : ; log S c º is used instead of log a, the integer entries of block matrices D j will be changed to rational entries because of identities (B), but the eigenvalues of D j will be unchanged. This means that, whenever the eigenvalues of D are concerned, we may assume that j = j C1 is torsion-free, or j D p j . Consequently, we may assume that GR. N j / D GR. N ' j /. Since p j = p j C1 Š Z k j , by taking the tensor product with R, it is known that GR. N ' j / D sp.D j /. Thus GR. N j / D sp.D j /. Now the theorem follows from Lemma 3.1.
Remark 3.4. From Theorem 3.3, it follows that the growth rate of any endomorphism on a finitely generated torsion-free nilpotent group is an algebraic integer. The question of determining groups for which the growth rate of a group endomorphism is an algebraic number was raised by R. Bowen in [2, p. 27]. In fact, we show in Theorem 3.7 that it is always the case that GR. / D sp.D 1 /.
We consider another example in which we obtain much information about linearizations of endomorphisms, and then we obtain an idea of proving the next result, Theorem 3.7. (i) For any ij 2 S, M i;j p;q is unique for which pq 2 S 2 . (ii) If ij 2 S S 2 , then ij is a word w of elements in S˙1 2 modulo 3 . If w ¤ 1 modulo 3 , the .i; j /-column of K is an integer combination of .p; q/-columns of K corresponding to the elements pq appearing in the word w. If w Á 1, then M i;j p;q D 0 for which pq 2 S 2 . (iii) The right-hand side of the expression (P) can be rewritten in terms of only the elements of S 2 using the words ij Á w. pq /. This yields the second block D 2 .
Since ij D OE 1i ; 1j , taking on both sides, we have (see [5,  The effect of part (iii) on K and hence on K 0 is doing some row operations using the .i; j /-rows in the last block of K 0 for which ij D w. pq / ¤ 1 modulo 3 . By rearranging S further to S 2 [ S 1 2 [ S 2 2 , we have The middle block column is determined by the fact that if ij Á w Á 1, then M i;j p;q D 0 for which pq 2 S 2 . Consequently, the second block D 2 of D is a block submatrix of K 00 which is obtained by removing the rows and columns associated to S S 2 . Note also that K; K 0 and K 00 have the same eigenvalues which contain the eigenvalues of D 2 . This observation shows that sp.D 1 / D max¹j i jº max¹ p i j º D sp.K/ 1=2 sp.D 2 / 1=2 : For the next inductive step, we recall that every element of S 3 . T 3 / is of the form OE 1`; ij , where i < j . Taking  This produces the matrix K D D 1˝V 2 D 1 . First if OE 1r ; pq D w.S 3 / ¤ 1 modulo 4 , by doing some column operations and then by doing some row operations, we obtain a matrix K 00 , which can be regarded as a lower triangular block matrix. Finally, we remove the columns and rows from K 00 which are associated with the elements OE 1r ; pq D w.S 3 / modulo 4 . This gives rise to the third block D 3 of D . Hence sp.D 3 / Ä sp.D 1 / 3 . Continuing in this way, we may assume that the j th block D j of D is obtained from N j 2 D 1 ˝V 2 D 1 so that sp.D 1 / sp.D j / 1=j : Consequently, GR. / D max¹sp.D j / 1=j º D sp.D 1 / D GR. ab / Ä sp.D /.