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In this paper we show that evolution algebras over any given field $\Bbbk$ are universally finite. In other words, given any finite group $G$, there exist infinitely many regular evolution algebras $X$ such that $Aut(X)\cong G$. The proof is built upon the construction of a covariant faithful functor from the category of finite simple (non oriented) graphs to the category of (finite dimensional) regular evolution algebras. Finally, we show that any constant finite algebraic affine group scheme $\mathbf{G}$ over $\Bbbk$ is isomorphic to the algebraic affine group scheme of automorphisms of a regular evolution algebra., Comment: Minor corrections. Bibliography updated. To appear in Proc. Amer. Math. Soc

The Steenrod problem asks: given a G-module, when does there exist a Moore space realizing the module? By using the equivariant Postnikov Tower, it is shown that a ZG-module is ZG-realizable if and only if it is ZHrealizable for all p-Sylow subgroups H, for all primes pl GI. Let G be a finite group. Let M be a finitely generated ZG-module. We say that M is a Steenrod representation if there exists a Moore space X with G-action such that the homology of X is isomorphic to M as a ZG-module. Recall that a Moore space is a topological space whose reduced homology vanishes in all dimensions except one. Without loss of generality, M will be assumed 2-free [1] from now on. In this paper, the following statement will be proved: M is a Steenrod representation as a ZG-module if and only if M is a Steenrod representation as a 7H-module for all p-Sylow subgroups H of G, for all primes pII GIC. This statement is inspired by papers by J. Arnold [1, 2] and P. Vogel [18]. Arnold [2] showed that if G is cyclic, then every finitely generated ZG-module is a Steenrod representation. And Vogel showed that if for any ZG-module M, we can always find a G-Moore space X realizes M, then G has only cyclic Sylow subgroups. Let us first consider a Moore space X whose homology is isomorphic to M as a 2-module. Let G(X) denote the space of self homotopy equivalences of X and let BG(X) [9] denote the classifying space of the H-space G(X) for a certain fibration defined by Dold and Lashof [10]; see also Stasheff [15]. Since a homotopy equivalence of X induces an automorphism on M, there is a map G(X) -* Aut(M). And the map G(X) -* Aut(M) induces another map a : BG(X) -' BAut(M). On the other hand, since M is a ZG-module, there is a map G -* Aut(M) (= GL(n, 2) if M = Zn), which induces .p : BG -) BAut(M). Following Cooke's theory [9], as pointed out in [12], M is a Steenrod representation as a ZG-module if and only if the lifting f exists in

Let F, G E C[x, y]. If the Jacobian determinant of F and G is 1, then G is said to be a Jacobian mate of F. If, in addition, G has degree less than that of F, then G is said to be a younger mate of F . In this paper, a necessary and sufficient condition is given for a polynomial to have a younger mate. This also gives rise to a formula for the younger mate if it exists. Furthermore, a conjecture concerning the existence of a younger mate is shown to be equivalent to the Jacobian conjecture. Throughout this paper, F and G will be polynomials in C[x, y] where C denotes the field of complex numbers. We say that F and G satisfy the Jacobian hypothesis if their Jacobian determinant is one, i.e., Fx Gy Fy Gx = 1 . In this case, we also say that G is a Jacobian mate of F. Furthermore, if the x-degree (resp. y-degree, total degree) of G is less than that of F, then G is said to be a younger mate of F relative to the x-degree (resp. y-degree, total degree). For instance, x + y has younger mates y and -x relative to the x-degree and the y-degree, respectively, but has no younger mate relative to the total degree. This paper was motivated by the Jacobian conjecture which asserts that if F has a Jacobian mate G, then (F, G) is an automorphism pair. In Section 1, it is shown that a younger mate is unique (up to an additive constant) and universal, i.e., if a Jacobian mate G of F exists, then any other mate of F can be expressed as G plus a polynomial in F. In Section 2, the problem of existence of a younger mate of F is reduced to the case where F is monic in both variables. In Section 3, a necessary and sufficient condition for the existence of a younger mate and a formula for a younger mate provided one exists are given. Finally, in Section 4, a conjecture concerning the existence of younger mates is formulated and shown to be equivalent to the Jacobian conjecture. Received by the editors July 6, 1993 and, in revised form, January 18, 1994. 1991 Mathematics Subject Classification. Primary 13B25, 13F20, 14E09, 16W20.

It is well known that a generic compact Riemann surface of genus greater than two admits only the identity automorphism; however, examples of such Riemann surfaces with their defining algebraic equations have not appeared in the literature. In this paper we give the defining equations of a doubly infinite, two-parameter family of projective curves (Riemann surfaces if defined over the complex numbers), whose members admit only the identity automorphism. It is well known that a generic curve of genus greater than two admits only the identity automorphism. Although this result was probably known by the turn of the century, the first published proof was given by Bailey in 1961 [1]. To obtain the strongest results, Bailey's method is necessarily nonconstructive; it does not yield an example of a defining algebraic equation for a curve with no nontrivial automorphisms. Similarly a proof by Greenberg [4], using techniques of Teichmuller theory, does not yield an explicit example of a Riemann surface with a trivial automorphism group. Much of the subsequent work on automorphisms of Riemann surfaces, including the author's, has relied on the representation of a given Riemann surface as the upper half plane under the action of a Fuchsian group. This again has the disadvantage of rarely yielding a defining algebraic equation for the given Riemann surface. Indeed, in the preface to his book, The complex analytic theory of Teichmiiller spaces, Subhashis Nag exclaimed, "Almost every compact Riemann surface of genus g > 3 allows only the identity automorphism. (I don't know, though, of even a single explicit such algebraic curve whose automorphism group is demonstrably trivial!)." The author believes that examples pertinent to famous theorems should be readily at hand. Therefore, in this paper we give the defining equations of a doubly infinite, two-parameter family of curves which admit only the identity automorphism (see equation (1) below). The curves in this family have genus (n 1)(m 1)/2 for relatively prime integers m and n which satisfy n > m+ 1 > 3. Let C be a curve defined by (1) and let C' be a nonsingular projective model for C. The proof that C' admits only the trivial automorphism depends on the Received by the editors February 5, 1993. 1991 Mathematics Subject Classification. Primary 14E09, 14H55, 30F99. @ 1994 American Mathematical Society 0002-9939/94 $1.00 + $.25 per page

We say that an automorphism Φ \Phi of E ∗ {\mathcal {E}^ \ast } (the lattice of recursively enumerable sets modulo the finite sets) is induced by a permutation p iff for all e, Φ ( W e ) = ∗ p ( W e ) \Phi ({W_e}){ = ^ \ast }p({W_e}) . A permutation h is called a presentation of Φ \Phi iff for all e, Φ ( W e ) = ∗ W h ( e ) \Phi ({W_e}){ = ^\ast }{W_{h(e)}} . In this paper, we will explore the degree-theoretic connections between these two notions. Using a new proof of the well-known fact that every automorphism is induced by a permutation p, we show that such a p can be found recursively in h ⊕ ∅ h \oplus \emptyset , where h is a presentation of Φ \Phi . The main result of the paper is to show that there is an effective automorphism of E ∗ {\mathcal {E}^ \ast } which is not induced by a Δ 2 {\Delta _2} -permutation.

The maximal nilpotent and supersoluble automorphism groups of Riemann surfaces were given in earlier papers by this author. In this note the author wishes to correct the necessity of the condition given in Theorem (4.3) of Bounds for the order of supersoluble automorphism groups of Riemann surfaces (Proc. Amer. Math. Soc. 108 (1990), 587-600), which was left out at the time of writing the paper. The author also wishes to apologize to the readers for that.

Let E -X be an elliptic modular surface and S the tangential ruled surface of a projective embedding of X. The divisor that collects the involutions of the elliptic fibers of E is precisely the branch locus of E -S (at least generically). In this paper, we present two theorems that characterize this divisor in terms of the action of the group of modular automorphisms. These results extend work of D. Burns [1]. 0. INTRODUCTION AND STATEMENTS OF THE MAIN RESULTS The purpose of this paper is to present two theorems that together extend a result of D. Burns [1, Theorem 2]. This extension gives a rather complete picture of the invariant-theoretical question addressed there. We believe that these results will prove useful in the construction of elliptic modular surfaces with prescribed finite group as automorphisms. Thus, suppose that F is a torsion-free subgroup of finite index in F(1) SL2 (Z), and let X = X(F) be the usual modular curve attached to F [cf. 3 for details on modular curves]. There is an elliptic surface E = E(F) fibered over X such that, whenever F < A < F(l), the finite group ' := A/F acts on E in a very nice way [cf. I or 4]. In particular, -1 E F(1) induces an involution of E, and the quotient S S/ (-1) is birationally ruled over X. In [1], Burns constructs intrinsically a minimal model S of S (at least for almost all F). This geometric model S is a ruled surface over X, and Burns shows [ 1, Theorem 1] that extrinsically S is the tangential ruled surface of any projective embedding of X. Hence, to construct E explicitly from X and ' one should pinpoint the branch locus of the map E -* S. One component of this branch locus is the zero-section Z c S. A second component is the image & of a smooth 3-fold section bE c E. The divisor class of & is D := 3Z + 3Kx + KrX, where a canonical divisor Kx and the Received by the editors March 28, 1989 and, in revised form, December 19, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 14J27; Secondary 144D25, 14J50. ? 1990 American Mathematical Society 0002-9939/90 $1.00 + $.25 per page

We show, in Zermelo-Fraenkel set theory without the Axiom of Choice, that the existence of a discontinuous homomorphism of the additive group of real numbers induces a selector for the Vitali equivalence relation $\mathbb{R}/\mathbb{Q}$. This shows that a nonprincipal ultrafilter on the integers is not sufficient to construct a discontinuous automorphism of the complex field, confirming a conjecture of Simon Thomas. This is an improved version of our paper in the Proceedings of the American Mathematical Society, which used a weak version of the Axiom of Choice for the same result.

Let $(X,D)$ be a log smooth pair of dimension $n$, where $D$ is a reduced effective divisor such that the log canonical divisor $K_X + D$ is pseudo-effective. Let $G$ be a connected algebraic subgroup of $\mathrm{Aut}(X,D)$. We show that $G$ is a semi-abelian variety of dimension $\le \min\{n-\bar{\kappa}(V), n\}$ with $V := X\setminus D$. In the dimension two, Shigeru Iitaka claimed in his 1979 Osaka J. Math. paper that $\dim G\le \bar{q}(V)$ for a log smooth surface pair with $\bar{\kappa}(V) = 0$ and $\bar{p}_g(V) = 1$. We (re)prove and generalize this classical result for all surfaces with $\bar{\kappa}=0$ without assuming Iitaka's classification of logarithmic Iitaka surfaces or logarithmic $K3$ surfaces., Comment: minor changes; to appear in Proceedings of the AMS

Recently, we introduced a space [Hfi(A);B] which consists of Banach space-valued distributions for which the Hankel transformation is an automorphism (The Hankel transformation of a Banach space-valued generalized function, Proc. Amer. Math. Soc. 119(1993), 153-163). One of the cornerstones in distribution theory is the kernel theorem of Schwartz which characterizes continuous bilinear functionals as kernel operators. The object of this paper is to prove a kernel theorem which states that for an arbitrary element of [Hf, x A ; B], it can be uniquely represented by an element of [H^A) ; B] and hence of [H^ ; [A ; B]]. This is motivated by a generalization of Zemanian (Realizability theory for continuous linear systems, Academic Press, New York, 1972) for the product space Dgn x V where F is a Frechet space. His work is based on the facts that the space D$n is an inductive limit space and the convolution product is well defined in Dr . This is not possible here since the space Hn(A) is not an inductive limit space. Furthermore, D(A) is not dense in Hf,(A). To overcome this, it is necessary to apply some results from our aforementioned paper. We close this paper with some applications to integral transformations by a suitable choice of A .

We prove several cases of the following theorem: Every free group word which is not a proper power can represent every permutation of an infinite set. The remaining cases will be proved in a forthcoming paper of R. C. Lyndon. Fx denotes a free group freely generated by the set X. The elements of X are called letters, and the elements of Fx are represented by reduced words in those letters. G denotes an arbitrary group. We say that a word w can represent a in G if a E G and there exists a homomorphism h: Fx G C such that h(w) = a. In other words w can represent a in G iff the equation w = a is solvable in G, where X is the set of unknowns. Notice that, if q is an endomorphism of Fx and +(w) can represent a in G, then w can represent a in G. As was pointed out in [3], the sums of exponents of each letter of w have a greatest common divisior 1 iff w can represent every a in every G. This set of words with g.c.d. 1 is larger than the set of words w for which there exists an automorphism a of Fx such that a(w) E X. E.g., by a theorem of J. H. C. Whitehead (see [5]), the words x2yx-1y-1 and x2y3, for x, y E X, are in the first set but not in the second set. The solvability of equations w = a for various groups G has been considered in several papers, see [2, 3, 4, 6, 7, 10]. In the present paper we will study this question for G = Sy, the group of all permutations of an infinite set Y. In [10] Silberger asks if every w E Fx which is not a power, i.e., is not of the form vk with k > 1, can represent every 7r in Sy, if Y is infinite. He shows that this is true if w = xmyn for x, y E X, x $ y and m $ 0 $ n. Another proof is given in [2]. (For finite Y sufficient conditions in terms of m and n were given in [4]). Our main result is the following partial solution of Silberger's problem (it was announced in [8]). THEOREM 1. If w E Fx, w is not a power, 7r E Sy, and among the cycles of 7r at least one cycle size appears infinitely many times (fixed points are counted as cycles), then w can represent ir in Sy. REMARK ADDED IN MARCH 1986. R. C. Lyndon has just removed my assumption about the existence of repeating cycles in 7r. Thus Silberger's conjecture is true. My proof uses some rather difficult tools of combinatorial group theory (a theorem of Weinbaum about subwords of a relator and the asphericity theorem for Cayley complexes of groups with one relator) but Lyndon's addition is more direct. First he points out that Theorem 1 reduces the problem to the cases when YI = No and either 7r has at least one infinite cycle or 7r has no infinite cycles but Received by the editors June 12, 1985 and, in revised form, October 1985 and March 1986. 1980 Mathematics Subject C(Lsification (1985 Revisnm). Primary 20F05; Secondary 20B07, 20E05. (?)1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page

We show that pairwise disjointness implies multiple disjointness for metric, weakly mixing regular minimal flows with an abelian phase group. A result on the disjointness of graphic minimal flows is also included. In a recent paper Auslander and Markley studied graphic flows and multiple disjointness for integer actions. Their major result is the following: Given a family of graphic minimal sets (Xi, Ti), Ti a homeomorphism of Xi, and nonzero integers a(i), if the flows (Xi, Ta(i)) are pairwise disjoint, then the product of all the flows is minimal. A related result is that, given a graphic flow (X, T), then (X, Ti) and (X, Tn) are disjoint whenever m :$ +/ n. In this paper we generalize the related result to Zn actions. We intended to generalize the major result to Rn actions, but found rather surprisingly that the result held for metric, weakly mixing regular minimal flows with abelian phase group, a substantially stronger result. The techniques used should be useful for proving generalizations of other results about graphic minimal flows as well as related results in [KN76] and [W75]. I would like to thank Ed Ihrig for useful conversations, Joe Auslander for catching an error in my original manuscript, and the referee for allowing me to include an improved result in the revised copy. Preliminaries. Standing assumptions. We assume that a flow (X, T) has a compact phase space and locally compact abelian phase group. Let J be the set of idempotents in the universal minimal set for T. DEFINITION. A flow (X, T) is a regular minimal flow iff it is a minimal flow such that, for any pair of points x, x' with (x, x') an almost periodic point in (X, T) x (X, T), there exists an automorphism h of (X, T) with h(x) = x'. DEFINITION. A minimal flow (X, T) is weakly mixing minimal iff it has no nontrivial equicontinuous factor. When X is metric this is equivalent to (X, T) x (X, T) having a point with dense orbit. DEFINITION. A flow (X, T) is totally minimal iff (X, S) is minimal for all syndetic subgroups S of T. DEFINITION. A flow (X, T) is graphic iff it is a weakly mixing minimal flow, and Xu is a single orbit for all idempotents u in J iff it is a minimal flow such that, for any pair of points x, x' with (x, x') an almost periodic point in (X, T) x (X, T), there exists a t in T with xt = x'. Note a graphic flow is a regular minimal flow and is totally minimal by 2.32 of [GH]. First we consider when pairwise disjointness implies multiple disjointness. Received by the editors October 10, 1984 and, in revised form, September 16, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 54H20. ?)1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page

Examples are constructed of compact 3-manifolds with boundary whose groups of self-homotopy-equivalences are not finitely-generated. For a finite CW-complex X, let &(X) denote the space of basepoint-preserving homotopy equivalences from X to X, and let C(X) denote the group of homotopy equivalences fr(d'(X)). An obvious question is: Under what conditions on X must C(X) be finitely-generated? Sullivan [7] and Wilkerson [10] showed that if X is simply-connected, then 9(X) is finitely-presented. For non-simply-connected complexes, however, 9(X) can be infinitely-generated (have no finite generating set) even for seemingly uncomplicated examples. Frank and Kahn [3] showed that C9(S1 V SP V S2P-1) is infinitely-generated when p >r 2, and in [5] the author gave infinitely many examples of finite four-dimensional K(vr, 1)-complexes with Aut(Qr) and, hence, 9( K(v, 1)) infinitely-generated. In [5], it was asked whether there was an example of an aspherical 2-complex X with C(X) infinitely-generated. Recently, such examples were found by Brunner and Ratcliffe [2]. These various examples show two distinct ways that 9( X) can fail to be finitely-generated. In the Frank and Kahn examples, r2p_ (Sl V Sp V S2p-1) is quite large-it is infinitely-generated as a Zg1(S1 V SP V S2p-l)-module-and many elements of l9(Sl V Sp V S2p-1) arise by mapping the S2p-1 using an element of g2p_1(S1 V Sp V S2p-1). In the aspherical examples, Aut(g'1(X)) is infinitely-generated and the asphericity forces 9(X) -> Aut(vr1(X)) to be surjective. Clearly, if C( X) -b Aut(gl ( X)) is surjective then so is ( X v S" ) Aut(rj ( X V S" )) for n >? 2, so one can produce nonaspherical examples in dimension two. What is not apparent from these examples is the answer to the following question posed in [2]. Question. Is there a finite two-dimensional complex X with Aut(v'g(X)) finitelygenerated but 9( X) not finitely-generated? Received by the editors July 29, 1983. This paper was presented June 21, 1982 at the Conference on Combinatorial Methods in Topology and Algebraic Geometry held at the University of Rochester. 1980 Mathematics Subject Classification. Primary 55P10, 57M99; Secondary 20H25. Kev wt,ords anid phrases. Homotopy equivalence, self-homotopy-equivalence, 3-manifold, 2-complex, automorphism group, group ring. I This research was supported in part by the National Science Foundation. ,("1984 American Mathematical Societv 0002-9939/84 $1.t)t) + $.25 per page 625 This content downloaded from 207.46.13.156 on Sat, 10 Sep 2016 04:36:21 UTC All use subject to http://about.jstor.org/terms 626 DARRYL Mc CULLOUGH For a 3-manifold M let M' denote the result of removing from M the interiors of two disjoint closed 3-balls tamely-imbedded in the interior of M. Our main result is THEOREM 1. Let M be a compact aspherical 3-manifold-with-boundary, such that Out(7r1(M)) is finite and 7r1(M) admits a surjective homomorphism onto Z X Z. Then V(M') is infinitely-generated. Many instances of Theorem 1 are given in COROLLARY. Let M be a compact orientable 3-manifold-with-boundary such that the interior of M admits a complete hyperbolic structure with finite volume, and such that r1(M) admits a surjective homomorphism onto Z X Z. Then V(M') is infinitelygenerated. PROOF. M is aspherical since its interior is, and Out( T1( A)) is well known to be finite [6, p. 116; 8, p. 5.31]. C1 For example, the (compact) complement of the Whitehead link and the (compact) complement of the Borromean rings are familiar 3-manifolds which satisfy the hypotheses of the theorem and the corollary. Since any compact 3-manifold-with-boundary has the homotopy type of a finite 2-complex, and Out(71(M')) finitely-generated implies Aut(71(M')) finitely-generated, Theorem 1 answers the question of Brunner and Ratcliffe in the affirmative. We will give the proof of Theorem 1 in ?1, making use of two auxiliary theorems. These theorems, which are of independent interest, are proved in ??2 and 3. I wish to thank Andy Miller for helpful discussions concerning Theorem 2(b). 1. Proof of Theorem 1. Write S for 71(M, *) _71(M', *). Let t': V(M') -> Aut(7) be the homomorphism defined by (((f )) = f4. Let 91(M') = P1-'({I}) and VInn(M') = (I-1(Inn(7")). Since Out(7") is finite, VInn(M') has finite index in 9(M'), so to prove the theorem it suffices to show VInn(M') is infinitely-generated. Let M1 = M if M is orientable, otherwise let M1 be the orientable double cover of M. Now M1 is compact, orientable, and has a boundary component which is not a 2-sphere. Therefore, H1(Ml; Z) is infinite so M1 is sufficiently large. Therefore, the center of 7"1(Ml) is finitely-generated [9]. This implies that the center of ST is finitely-generated. Using Theorem 2(b), which will be stated and proved in ?2, we see that WInn(M') is infinitely-generated if g9 (M') is. Let Aut,(72(M')) be the group of n-module automorphisms of 72(M'). We will prove that the natural homomorphism 91(M') -> Aut ,(7T2(M')) is surjective. Let K be a finite 2-complex having the homotopy type of M; then K is aspherical and K' = K V S2 V S2 has the homotopy type of M'. Since S has cohomological dimension two, the k-invariant k(K') is zero. As shown in [2], this implies 91(K') -Aut,(7"2(K')) is surjective. (This surjectivity can be proved directly for K' without difficulty: just define a homotopy equivalence that is the identity on K and induces the desired automorphism on 7"2(K').) Therefore, 9,(M') -Aut,(7T2(M')) is surjective, so 91(M') is infinitely-generated if Aut,(72(M')) is. But 72(M') _r 72(K') _ Z7 E Z7, so Aut,(72(M')) _ GL2(Z7), the group of 2 x 2 invertible matrices This content downloaded from 207.46.13.156 on Sat, 10 Sep 2016 04:36:21 UTC All use subject to http://about.jstor.org/terms COMPACT 3-MANIFOLDS 627 with entries in Zv. We apply Theorem 3, which will be stated and proved in ?3, with G = 7 to show that GL2(Z7) is infinitely-generated. This completes the proof of Theorem 1. C} 2. Proof of Theorem 2. While part (a) of Theorem 2 is not needed in the proof of Theorem 1, it is of independent interest and can be proved without much more work than that needed for part (b). A special case of part (a) appears in [4]. THEOREM 2. Let L be a finite-dimensional locally-finite connected simplicial complex. Then: (a) If 7r1(L, *) is centerless, then l00nn(L) V1(L) X 7T1(L, *). (b) If the center of T, (L, *) is finitely-generated, then $lnn( L) is infinitely-generated if and only if 91 (L) is infinitely-generated. PROOF. Write 7 for 7rT(L, *). Replacing L by the interior of a regular neighborhood of L, we may assume L is a triangulated open manifold, and the basepoint * is a vertex of L. Let N be a regular neighborhood in L of the 1-skeleton of L. Define a: 7T > lnn(L) as follows. For each a E 7T, choose an isotopy Ha: L x I -* L starting at the identity map 1, so that the trace of Ha (the homotopy class of the restriction of HO to * x I) equals a-', and so that the restriction of Ha to (L int(N)) x {t} equals the identity for all t E I. Let ha(x) = H0(x, 1). Note that for T GE T, (h0)#(T) = UTU1, so we can define a(a) = (ha). We will now show that a is a homomorphism. For homotopies G, H: L X I -* L with G(x, 1) = H(x, 0), we define (G * H)(x, t) to be G(x, 2t) if 0 Z X Z. Then GL,(ZG) is infinitely-generated. This content downloaded from 207.46.13.156 on Sat, 10 Sep 2016 04:36:21 UTC All use subject to http://about.jstor.org/terms 628 DARRYL Mc CULLOUGH PROOF. Let s and t be generators of Z x Z, and denote the group ring Z[Z x Z] = Z[s, s5-, t, t -] by R. The homomorphism -q induces a homomorphism ,B: GL2(ZG) GL2(R). Let S = SL2(R) n im(/3). We begin by using an idea from [2] to show that GL2(ZG) is infinitely-generated if S is. There is a short exact sequence det 1 -SL2(R) -GL2(R) R* 1 where the group of units R* is generated by {-1, s, t }. Let Ro c R* be the subgroup generated by {S2, t2}, which has index 8 in R*, and let H = det-1(RO). If w c Ro then the "positive" square root w1/2 is uniquely defined, and f: H -SL2(R) defined by f(A) = (det(A))-1/2A is a retraction. Let K = image(/3). Since H has finite index in GL2(R), K n H has finite index in K. But fIK n H retracts K (n H onto S. Therefore, if S is infinitely-generated, then so is GL2(ZG). The proof that S is infinitely-generated is a minor modification (for the case P = Z) of the argument of ?2 of t1], and we use the notation of that paper. Choose x, y E G with (x) = s and (y) t. Lemma 1 of [1] is replaced by LEMMA 1'. E2(R) is contained in (S n SL2(Z[S, s-1, t])) *v (S SL2(Z[s, s-', ti ] where V is the intersection of the factors. LEMMA 2 is replaced by LEMMA 2'. Let S be a nonunit element of Z. Then, the matrices

It is our purpose in this paper to initiate a study of the algebraic properties of a left loop Q ( ⋅ ) Q( \cdot ) satisfying the identical relation (1) y ( z ⋅ y x ) = ( y ⋅ z y ) x \begin{equation} \tag {1} y(z \cdot yx) = (y \cdot zy)x \end{equation} for all x , y , z ∈ Q x,\;y,\;z \in Q . It is shown that (1) implies right division in Q ( ⋅ ) Q( \cdot ) . By introducing a new operation ’ ∘ \circ ’ in Q Q , the connection between the left loop Q ( ⋅ ) Q( \cdot ) and Bol loop Q ( ∘ ) Q( \circ ) is established. Further we show that the role of nuclei in the left loop theory is not the same as that in the loop theory. We conclude the paper by describing situations in which the left loop Q ( ⋅ ) Q( \cdot ) is Moufang.

We describe a construction which yields unbounded derivations and strongly continuous one-parameter automorphism groups of certain group C*-algebras. As an application, we show that a simple C*-algebra can have an automorphism group which is not approximately inner. 1. A strongly continuous one-parameter automorphism group of a C*algebra 21 is a continuous homomorphism t h> a, from the real numbers into the group of *-automorphisms of 21 equipped with the topology of simple norm convergence. We say that (at) is approximately inner if there is a sequence (Hn) of selfadjoint elements of 21 such that \\e'lH"Ae~"H" at(A)\\ -*'0 uniformly on compact subsets of R, for each A in 21. Strongly continuous one-parameter automorphism groups of UHF algebras have been investigated in a recent series of papers by Powers and Sakai [5]-[8], who conjecture that such groups are always approximately inner. In this paper we show how to construct strongly continuous automorphism groups in certain group C*-algebras which are not approximately inner. In particular, the construction works when the group is the free group on two generators. The reduced C*-algebra of this group is simple [4], so our results show that a simple C* -algebra with identity can, in general, have strongly continuous one-parameter automorphism groups which are not approximately inner, in contrast to the Powers-Sakai conjecture for the UHF case. We should like to thank R. Powers and S. Sakai for supplying us with preprints of their work. 2. Let G be a (discrete) group. For h in G let eh be the characteristic function of {h}, so that {eh: h G G) is an orthonormal basis for / (G). For g in G define a unitary operator U on l2(G) by U eh = e h. Let 2l0 be the set of all finite linear combinations of elements of {U : g G G}. The reduced C*-algebra C* (G) of G is the norm closure of 2t0 in the algebra of all bounded operators on I (G). The (full) C*-algebra C*(G) of G is the enveloping C*-algebra (see §2.7 of [2]) of the convolution group algebra lx(G). Define a state w of C* (G) by Received by the editors March 16, 1976. AMS (MOS) subject classifications (1970). Primary 46L05. 1 Research supported by a grant from MOSOI, Iran. ©American Mathematical Society 1977 310 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use UNBOUNDED DERIVATIONS OF GROUP C*-ALGEBRAS 311

In this short paper we give a completely elementary argument for the Thompson-Wielandt theorem and note its significance for the GoldschmidtSims conjecture. A recent fundamental paper of Goldschmidt [Gsl] has renewed interest in the study of amalgams of finite groups. The underlying problem of the current works on amalgams is the so-called Goldschmidt-Sims problem. A theoretical starting point for this is the Thompson-Wielandt theorem [T]. This asserts that every "nontrivial" primitive amalgam must be of characteristic p-type and that the common group in the amalgam is "almost" a p-group, for some prime p. Here, we offer a transparent argument of how the Thompson-Wielandt theorem is a direct consequence of primitivity. Please consult [Gsl] for the definitions not given below. We denote an amalgam by Pi D B C P2, where P1, P2 and B are the obvious subgroups of P1 *B P2. It is said to be primitive if, for i = 1, 2, whenever 1 74 K a Pi and K C B, we have Np3, (K) = B. A justification for primitive amalgams is that any two distinct maximal subgroups of a finite simple group form such an amalgam. Further, the completion graph of the amalgam given by two minimal parabolics of a rank 2 Lie-type group is the building associated to that group. The rather ambitious Goldschmidt-Sims problem is then to classify all possible primitive amalgams. Prompted by possible applications, many special cases of the above have been done. Among these are the works of Goldschmidt [Gsl], Delgado [D], Fan [F], Stellmacher [Stl], and Stroth [Strl, Str2]. It was Goldschmidt who first fully and successfully exploited the underlying geometry of amalgams. Take P1 D B C P2 to be an arbitrary primitive amalgam, set G = P1 *B P2, and let F be the usual coset graph, i.e., the completion graph F(G, P1, P2) (see [Gsl]). It is then well known that (1) F is a tree whose valences at adjacent vertices are lPi: BI and JP2: Bl. (2) G is an edge transitive group of automorphisms of F. (3) For adjacent vertices a and d in F, G0, D Gc,,fl C G,g is isomorphic to P1 D B C P2. It has been usually useful to investigate a baby version of the Goldschmidt-Sims problem, which we refer to as the Goldschmidt-Sims conjecture: There exists an integer constant M such that for any G and F as described above, no nontrivial element of G fixes all the vertices with distance M from some vertex a in F. Under an additional stipulation, Goldschmidt [Gs2] has obtained a (nongeometric) bound on the number of possible primitive amalgams with a given index. Of Received by the editors May 22, 1984 and, in revised form, June 1, 1985. 1980 Mathematics Subject Classification. Primary 20D05; Secondary 20B05. (?)1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page

In his paper, L. Hörmander has used the Weyl calculus to study the Fourier integral operator theory. In the present paper, the author considers the correspondences W τ {W_\tau } , τ ∈ R \tau \in R ( R R is the set of the real numbers), which mean the standard correspondence of symbol and operator if τ = 0 \tau = 0 , and the correspondence of Weyl type if τ = 1 / 2 \tau = 1/2 , and shows the explicit asymptotic formula which describes the deviation of W σ ( W τ ) − 1 {W_\sigma }{\left ( {{W_\tau }} \right )^{ - 1}} from the automorphisms as Lie algebra, and makes some remarks on the above formula.

In this paper, we extend a result of Joseph and Finkelstein and show that there is no group G such that G' is an extra-special p-group of exponent > p (p odd). K. Joseph and L. Finkelstein [2] have shown that if p is an odd prime, there does not exist a finite group G satisfying the following three conditions: (i) G' is an extra-special p-group of exponent > p. (ii) Z (G ) C G'. (iii) G acts irreducibly on G'/Z(G'). It is the object of this paper to prove that their result remains valid even if conditions (ii) and (iii) are dropped. That is, there is no finite group G such that G' is an extra-special p-group of exponent > p (p odd). Recall that a finite p-group G is called extra-special if Z(G) = G', and IG'l = p. We now list a series of lemmas which will be needed for the main theorem. In all that follows, p is an odd prime. LEMMA 1. Let G be an extra-special p-group. Then (a) (xy)P = xPyP, (b)xP E Z(G)for all x,y E G. PROOF. See [1, p. 183]. LEMMA 2. Let G be an extra-special p-group of exponent > p, and let U = {x E GlxP = 1}. Then U is a characteristic subgroup of G and [G: U] = P. PROOF. The fact that U is a characteristic subgroup follows immediately from Lemma l(a), and the fact that automorphisms preserve order. The map x -* xP is a homomorphism of G onto Z(G) with kernel U. Hence G/U Z(G) and [G: U] = p. LEMMA 3. Let G be a finite p-group of linear transformations acting on a vector space V over a field F of characteristic p. Then some nonzero vector of V is fixed by every element of G. PROOF. See [ 1, p. 3 1]. LEMMA 4. Suppose an abelian group G acts as a group of linear transformations Received by the editors August 15, 1975. AMS (MOS) subject classifications (1970). Primary 20F35. ?D American Mathematical Society 1976

Let Y be an Enriques variety of complex dimension 2n - 2 with n >= 2. Assume that n = 2m for odd prime m. In this paper we show that Y is the quotient of a product of a Calabi-Yau manifold of dimension 2m and an irreducible holomorphic symplectic manifold of dimension 2m - 2 by an automorphism of order n acting freely. We also show that both Y and its universal cover are always projective.

Let H H be a complex Hilbert space. The symbol A # B A\# B stands for the geometric mean of the positive bounded linear operators A , B A,B on H H in the sense of Ando. In this paper we describe the general form of all automorphisms of the set of positive operators with respect to the operation # \# . We prove that if dim ⁡ H ≥ 2 \dim H\geq 2 , any such transformation is implemented by an invertible bounded linear or conjugate-linear operator on H H .

An abelian p-group G is called (fully) transitive if for all x, y E G with U G (x) = U G (y) (U G (x) < U G (y)) there exists an automorphism (endomorphism) of G which maps x onto y. It is a long-standing problem of A. L. S. Corner whether there exist non-transitive but fully transitive p-groups with finite first Ulm subgroup. In this paper we restrict ourselves to p-groups of type A, this is to say p-groups satisfying Aut(G) |p ω G = U(End(G) |p ω G). We show that the answer to Corner's question is no if pG is finite and G is of type A.

This paper deals with analytic and geometric properties of the Maskit embedding of the Teichmüller space of once punctured tori. We show that the image of this embedding has an inward-pointing cusp and study the boundary behavior of conformal automorphisms. These results are proved using Y.N. Minsky’s Pivot Theorem.

Let D D be a division ring and let V = D n V=D^n be a finite-dimensional right D D -vector space, viewed multiplicatively. If G = D ∙ G=D^\bullet is the multiplicative group of D D , then G G acts on V V and hence on any group algebra K [ V ] K[V] . In this paper, we completely describe the semiprime G G -stable ideals of K [ V ] K[V] , and conclude that these ideals satisfy the ascending chain condition. As it turns out, this result follows fairly easily from the corresponding results for the field of rational numbers (due to Brookes and Evans) and for infinite locally-finite fields (handled in Part I).

Given the disk algebra A(ID) and an automorphism ae, there is associated a non-self-adjoint operator algebra 2+ x ( A(ID) called the semicrossed product of A(ID) with ae. Buske and Peters showed that there is a one-to-one correspondence between the contractive Hilbert modules XH over Z+ x, A(ID) and pairs of contractions S and T on XH satisfying TS = Sa(T). In this paper, we show that the orthogonally projective and Shilov Hilbert modules Xover 2+ x A(ID) correspond to pairs of isometries on XH satisfying TS = Sa(T). The problem of commutant lifting for 2+ x ( A(ID) is left open, but some related results are presented.

An I-E group is a group G in which every endomorphism is finitely generated by its inner automorphisms. In this paper a characterization for a semidirect sum of I-E groups to be an I-E group is obtained and some wellknown results are generalized. We then use this characterization to prove that a semidirect sum of finite I-E groups will again be an I-E group if the normal semidirect summand is unique and fully invariant. Conditions for a group to be an I-E group are also given.

Let N be an imaginary abelian number field. We know that h-, the relative class number of N, goes to infinity as fN, the conductor of N, approaches infinity, so that there are only finitely many imaginary abelian number fields with given relative class number. First of all, we have found all imaginary abelian number fields with relative class number one: there are exactly 302 such fields. It is known that there are only finitely many CMfields N with cyclic ideal class groups of 2-power orders such that the complex conjugation is the square of some automorphism of N. Second, we have proved in this paper that there are exactly 48 such fields.

Using the fundamental group of a punctured torus, a free group F of rank two, and the fact that the natural eipmorphism from AutF onto Aut(F/F') has as kernel the group of inner automorphisms of F, we describe representatives of the conjugacy classes of generating pairs of F and give explicit relations between them. Let F = F(S, T) be the free group on S and T. By a theorem of Nielsen [N] (see [LS, p. 25]) the natural epimorphism from AutF onto Aut(F/F') ( = GL(2, Z)) has as kernel the group of inner automorphisms of F. From this it follows easily that, if a is the abelianization homomorphism from F onto F/F' (= Z2) and a E 22 is primitivel, then the inverse image of a under a is a conjugacy class of primitive elements. Also, if (ai,a2) is a basis of 22, then, up to conjugacy, there is a unique basis (fi,f2) of F such that (fi)a = ai (i = 1,2). (The basis (fi, f2) is conjugate to (gl, 2) if there exists w E F such that w-lfiw = gi (i = 1,2)). In the important paper [OZ], Osborne and Zieschang define explicitly primitive words Wm,n E F(S,T), where m and n are relatively prime integers, such that (Wm,n)a = (m, n). They also state that if mn pq = 1, then (Wm,n, Wp,q) is a basis of F; this, while correct for nonnegative values of m,n,p, q, is not valid in general (for example W-2,-3 and W1,1 do not generate F). A composition formula is also stated in [OZ, Thm. 3.5] but this, even with the correction of indices in [LTZ, 2.1.3], is incorrect in general. In the present article we consider elements Va of F for a = (m, n) E Z2 and e E D C IR2 where D is the complement of the union of all the lines that intersect 22 in more than one point. If gcd(m,n) = 1, then V(,) is conjugate to Wm,n. We show in Theorem 1.i) that (Va, Vb) is a basis of F, if Z2 = (a, b), and obtain in Theorem l.ii) a composition formula. Everything is obtained by applying the fundamental group functor ir to the punctured torus. Denote by T the torus IR2/ 22, by To the punctured torus (R2 _ Z2) / Z2 and by p : R2 Z2 -To the natural projection. If a E 22 and e E D, then denote (e)p by E and define ya E 7r(To, ) as the homotopy class of the loop (e+ta)p, t E [0,1]. Denote ' o0) (resp. y0 ,1)) by S (resp. Te). There is an isomorphism Received by the editors September 15, 1997. 1991 Mathematics Subject Classification. Primary 57M07, 20E05.

For all g ≥ 2 g \ge 2 there is a Riemann surface of genus g g whose automorphism group has order 8 g + 8 8g+8 , establishing a lower bound for the possible orders of automorphism groups of Riemann surfaces. Accola and Maclachlan established the existence of such surfaces; we shall call them Accola-Maclachlan surfaces. Later Kulkarni proved that for sufficiently large g g the Accola-Maclachlan surface was unique for g = 0 , 1 , 2 mod 4 g= 0,1,2\mod 4 and produced exactly one additional surface (the Kulkarni surface) for g = 3 mod 4 g= 3\mod 4 . In this paper we determine the symmetries of these special surfaces, computing the number of ovals and the separability of the symmetries. The results are then applied to classify the real forms of these complex algebraic curves. Explicit equations of these real forms of Accola-Maclachlan surfaces are given in all but one case.

In this paper, we prove the following two results which generalize the theorem concerning automorphic-differential endomorphisms asserted by J. Bergen. Let R R be a ring, R F R _{\mathcal {F}} its left Martindale quotient ring and A \mathfrak {A} a right ideal of R R having no nonzero left annihilator. (1) Let C C be a pointed coalgebra which measures R R such that the group-like elements of C C act as automorphisms of R R . If R R is prime and ξ ⋅ A = 0 \xi \cdot \mathfrak {A}=0 for ξ ∈ R # C \xi \in R\#C , then ξ ⋅ R = 0 \xi \cdot R=0 . Furthermore, if the action of C C extends to R F R _{\mathcal {F}} and if ξ ∈ R F # C \xi \in R _{\mathcal {F}}\#C such that ξ ⋅ A = 0 \xi \cdot \mathfrak {A}=0 , then ξ ⋅ R F = 0 \xi \cdot R _{\mathcal {F}}=0 . (2) Let f f be an endomorphism of R F R _{\mathcal {F}} given as a sum of composition maps of left multiplications, right multiplications, automorphisms and skew-derivations. If R R is semiprime and f ( A ) = 0 f(\mathfrak {A})=0 , then f ( R ) = 0 f(R)=0 .

We construct an invertible disjointness preserving operator T on a normed lattice such that T−1 is not disjointness preserving. Two elements x1, x2 in an (Archimedean) vector lattice X are said to be disjoint (in symbols: x1 ⊥ x2) if |x1| ∧ |x2| = 0. Recall that a (linear) operator T : X → Y between vector lattices is said to be disjointness preserving if T sends any two disjoint elements in X to disjoint elements in Y, that is, Tx1 ⊥ Tx2 in Y whenever x1 ⊥ x2 in X . Assume now that a disjointness preserving operator T : X → Y is bijective, i.e., one-to-one and onto, so that T−1 : Y → X exists. It was conjectured by the first named author (see, for example, the problem section in [HL, page 143]) that T−1 is also disjointness preserving. The purpose of this note is to give a counterexample to this conjecture. Moreover, we will construct a bijective disjointness preserving automorphism T on a normed lattice G such that T−1 is not disjointness preserving. We precede our construction by several comments on some recent developments regarding the above conjecture. The two most important classes of vector lattices are the Banach lattices and the Dedekind complete vector lattices. The vector lattice G that we will construct is in neither of these classes. It cannot be a Banach lattice in view of the fact that, as has been recently shown by Huijsmans and de Pagter [HP] and independently by Koldunov [K], the above conjecture has an affirmative solution for Banach lattices. More precisely, they proved the following theorem. Theorem 1 ([HP, Theorem 2.1], [K, Theorem 3.6]). Let T be a one-to-one disjointness preserving operator from a relatively uniformly complete vector lattice X into a normed lattice Y . Then T is a d-isomorphism, that is, x1 ⊥ x2 in X if and only if Tx1 ⊥ Tx2 in Y . In particular, if TX = Y then T−1 : Y → X is a disjointness preserving operator. Since each Banach lattice is relatively uniformly complete, the previous theorem includes the case of Banach lattices. Here we should also cite the paper by K. Jarosz [J], which preceded [HP] and [K] and in which a special case of Theorem 1 was Received by the editors November 7, 1996. 1991 Mathematics Subject Classification. Primary 47B60.

Let X denote a Riemann surface which possesses a fixed point free group of automorphisms with a hyperelliptic orbit space. A criterion is proved which determines whether the hyperelliptic involution lifts to an automorphism of X. Necessary and sufficient conditions are stated which determine when a lift of the hyperelliptic involution is fixed point free. A complete determination is made of the abelian groups which may arise as automorphism groups of surfaces which possess a fixed point free lift. Hyperelliptic Riemann surfaces are natural objects of interest and have been studied quite extensively. They have simple defining equations and, since they admit an involution, they constitute a family of Riemann surfaces whose members admit a nontrivial automorphism. Covering surfaces of hyperelliptic surfaces have also been closely examined. The question of when a hyperelliptic surface can have a hyperelliptic cover was investigated in [2], [5], [8], and [10]. It was shown in [6] that if a Riemann surface X admits an abelian, fixed point free automorphism group H, then the hyperelliptic involution lifts to X. In addition, in [6] it was stated that if H is cyclic of prime order, then the lift of the hyperelliptic involution is never fixed point free. Further results concerning when the hyperelliptic involution lifts to a covering surface are contained in [1]. Let X be a compact Riemann surface which admits a fixed point free automorphism group H < Aut(X) with a hyperelliptic orbit space. In this paper we give necessary and sufficient conditions which determine when the hyperelliptic involution lifts to X. In addition, we give necessary and sufficient conditions which determine when such a lift is fixed point free. A complete determination is made of the abelian groups which may arise as automorphism groups of surfaces which possess a fixed point free lift. These results are combined with results in [12] to yield specific examples of automorphism groups yielding a hyperelliptic orbit space.

In this paper we present an example of an anticonformal automorphism whose square has prime order and is not embeddable. We prove that every embeddable automorphism of odd order of a compact Riemann surface is the square of an orientation-reversing self-homeomorphism. Finally we study whether a conformal involution is the square of an orientation-reversing automorphism.

An I-E group is a group G in which the endomorphism near-ring generated by the inner automorphisms of G equals the endomorphism near-ring generated by the endomorphisms of G. In this paper we obtain a result characterizing when a semidirect product of I-E groups of relatively prime orders is an I-E group. We then use this result to show that a semidirect product of cyclic groups of relatively prime orders is an I-E group.

Let V be an n × n n \times n unitary matrix. Then V induces an automorphism O V {O_V} of O ∞ {O_\infty } and O n {O_n} . It is shown in this paper that O V {O_V} is weakly inner. Let F n {F_n} be the UHF C ∗ {C^ \ast } -subalgebra of O n {O_n} and U be a unitary operator in the diagonal maximal abelian ∗ \ast -subalgebra of F n {F_n} . Then the automorphism λ U {\lambda _U} of On defined by λ U ( S i ) = U ∗ S i {\lambda _U}({S_i}) = {U^ \ast }{S_i} , for each generator S i {S_i} of O n {O_n} , is weakly inner. Any automorphism conjugate to either one mentioned above is also weakly inner.

In the group of continuous automorphisms of the field of Laurent series in one variable over a field of characteristic p > 0 , Sen's Theorem de- scribes the rapidity of convergence to the identity of the sequence formed by taking successive pth powers of a given element. This paper gives a short proof of Sen's Theorem, utilizing the methods of p-adic analysis in characteristic zero. The theorem in question appears in Sen's thesis (Sen), and is concerned with the group .%_ i (k) of formal power series in one variable with no constant term, and first degree coefficient equal to 1, over a field k of characteristic p > 0, where the group law is composition of series. If we call the variable t, this group is a closed subset of the discrete valuation ring K((t)), namely, the set of all u(t) for which u = t (mod?2). For the (Z)-adic filtration of group %,x, the successive quotients are isomorphic to the additive group k . Thus if we call u°" the «-fold iteration of u with itself, any time that u = t (mod/"), we necessarily have u°p = t (modz"+1). Sen's Theorem says much more and is best stated in terms of the additive valuation v of K((t)) normalized so that v(t) = 1. According to the theorem, if u°p" is not the identity, then v(u°p"(t)-t) = v(uop"\t)-t) (rnodp"). Let us abbreviate notation by setting iu(n) = i(n) := v(u°p"(t) - t). Sen's Theorem now says that if u°p" is not the identity, then i(n) = i(n - 1) (mod/?"). As examples of this phenomenon, we have, in characteristic 2, if u(t) = t + t4, then iu(n) = 22"+l ; if u(t) = t + t4 + /5, then iu(n) = 2n+2 ; and if u(t) = t + t3, then iu(n) = 1 + 2"+1 . It is easy to see why the first two of these facts hold, since each of t + t4 and t + Z4 +15 is an endomorphism of a formal group, and since in a formal-group endomorphism ring, the multiplication comes from substitution of power series. The first-mentioned series is an endomorphism of the additive formal group sf(x,y) = x + y, whose endomorphism ring has characteristic 2, and in that ring t + tA is g = 1 + tj>

This paper gives a proof of the theorem that for Coxeter groups the algebra of coinvariants is isomorphic to the normalizer of the Coxeter group in the linear group of a vector space over R \mathbb {R} . I have tried to give a relatively elementary proof which requires only elementary algebra and a little knowledge of the theory of invariants.