profinite completions of some groups acting on trees abstract we investigate the profinite completions of a certain family of groups acting on trees it turns out that for some of the groups considered the completions coincide with the closures of the groups in the full group of tree automorphisms however we introduce an infinite series of groups for which that is not so and describe the kernels of natural homomorphisms of the profinite completions onto the aforementioned closures of respective groups

hausdorff dimension pro p groups and kac moody algebras every finitely generated profinite group can be given the structure of a metric space and as such it has a well defined hausdorff dimension function in this paper we study hausdorff dimension of closed subgroups of finitely generated pro p groups g we prove that if g is p adic analytic and h c g is a closed subgroup then the hausdorff dimension of h is dimh dimg where the dimensions are of h and g as lie groups letting the spectrum spec g of g denote the set of hausdorff dimensions of closed subgroups of g it follows that the spectrum of p adic analytic groups is finite and consists of rational numbers we then consider some non p adic analytic groups g and study their spectrum in particular we investigate the maximal hausdorff dimension of nonopen subgroups of g and show that it is equal to 1 1 d 1 in the case of g sld fp t where p 2 and to 1 2 if g is the so called nottingham group where p 5 we also determine the spectrum of sl2 fp t p 2 completely showing that it is equal to 0 2 3 1 some of the proofs rely on the description of maximal graded subalgebras of kac moody algebras recently obtained by the authors in joint work with e i zelmanov

on the burnside problem for periodic groups narain gupta 1 and said sidki 2 1 university of manitoba department of mathematics winnipeg manitoba r3t 2n2 canada 2 university of brasilia department of mathematics brasilia d f brazil introduction the generalized burnside problem refers to the question are finitely generated periodic groups finite this was answered in the negative by golod 1 who proved that for each prime p there exists a finitely generated infinite p group golod s construction in not however direct and is based on his celebrated work with safarevi recently grigor uk 2 has given a direct and elegant construction of an infinite 2 group which is generated by three elements of order 2 in this paper we give for each odd prime p a direct construction of an infinite p group on two generators each of order p our group is a subgroup of the automorphism group of a regular tree of degree p and as might be expected it is residually finite and has infinite exponent preliminaries let p be an odd prime and let t 0 be the infinite regular tree of degree p with vertex 0 so that through each vertex u of t 0 there are p regular subtrees

equivariant stable homotopy and segal s burnside ring conjecture analogous results were proved later for ko in the generality of compact lie groups by atiyah and segal 8 and for kfq the algebraic k theory spectrum associated to the finite field fq by rector 29 using quillen s 26 computation of it kfq in each case the answer involves an appropriate completed representation ring of g and the cohomology theory in question is constructed from the permutative category of finite dimensional vector spaces over a field see 31 if one considers cohomology theories constructed from other permutative categories one expects to find analogous computations in terms of a completed representation ring of g in the given category appropriately defined in particular stable cohomotopy ts is constructed from the category of finite sets 31 and for this category the analogue of the representation ring is a well known object the burnside ring a g 13 a g is a commutative ring with augmentation so one may speak of a g the completed burnside ring moreover there is

the negative answer to kameko s conjecture on the hit problem abstract the goal of this paper is to give a negative answer to kameko s conjecture on the hit problem stating that the cardinal of a minimal set of generators for the polynomial algebra p k considered as a module over the steenrod algebra a is dominated by an explicit quantity depending on the number of the polynomial algebra s variables k the conjecture was shown by kameko himself for k 3 in his phd thesis in the johns hopkins university in 1990 and recently proved by us and kameko for k 4 however we claim that it turns out to be wrong for any k 4 in order to deny kameko s conjecture we study a minimal set of generators for a module p k in some so call generic degrees what we mean by generic degrees is a bit different from that of other authors in the fields such as crabb hubbuck nam repka selick wood we prove an inductive formula for the cardinal of the minimal set of generators in these generic degrees when the number of the variables k increases as an immediate consequence of this inductive formula we recognize that kameko s conjecture is no longer true for any k 4

on the behavior of the algebraic transfer let trk f 2 ph i bv k ext k k i a f 2 f 2 be the alge gl k braic transfer which is defined by w singer as an algebraic version of the geometrical transfer tr k s bv k s s 0 it has been shown that the algebraic transfer is highly nontrivial and more precisely that tr k is an isomorphism for k 1 2 3 however singer showed that tr 5 is not an epimorphism in this paper we prove that tr 4 does not detect the nonzero element g s ext 4 12 2s a f 2 f 2 for every s 1 as a consequence the localized sq 0 1 tr 4 given by inverting the squaring operation sq is not an epimorphism this gives a negative answer to a prediction by minami

the cohomology of the steenrod algebra and the mod p lannes zarati homomorphism abstract for any pointed space x the mod p lannes zarati homomorphism s h x is considered as a graded associated version of the mod p hurewicz map h q x h q x in the e 2 term of the adams spectral sequence in this paper we investigate the behavior of 3 f p and s h b z p s 1 for an odd prime p

dimension and randomness in groups acting on rooted trees we explore the structure of the p adic automorphism group of the infinite rooted regular tree we determine the asymptotic order of a typical element answering an old question of turan we initiate the study of a general dimension theory of groups acting on rooted trees we describe the relationship between dimension and other properties of groups such as solvability existence of dense free subgroups and the normal subgroup structure we show that subgroups of generated by three random elements are full dimensional and that there exist finitely generated subgroups of arbitrary dimension specifically our results solve an open problem of shalev and answer a question of sidki

equivariant stable homotopy and sullivan s conjecture let g be a p group and let x be a g complex let eg denote a contractible space on which g acts freely by the homotopy fixed point set of x we mean the fixed point set f eg x g where f eg x denotes the function space of all maps eg x equipped with a g action by 9f 9f9 1 if we let denote the one point space with trivial g action we may also consider the g space f x it is canonically g homeomorphic to x the g map e g induces a map x g f x g in 19 d sullivan proposed the following

on a 2 generated infinite 3 group subgroups and automorphisms this is the second paper dedicated to a detailed study of the infinite 3 group j 74 r 1 3 rr 3 1 y y ct 2 l a recursive definition to be explained later which was introduced by the author and narain gupta in 2 as a group of automorphisms of an infinite regular tree f as was shown in 4 y has the remarkable properties of containing a copy of every finite 3 group and having all its proper quotients finite the first paper s in this detailed study 99 dealt with the presentation problem here we study further aspects of its subgroup structure our prime objective though is to determine the group of automorphisms of 9 this task represents a large portion of our present paper the main results are as follows

on the peterson hit problem let pk f2 x1 x2 xk pk f2 x1 x2 xk be the polynomial algebra over the prime field of two elements f2f2 in k variables x1 x2 xkx1 x2 xk each of degree 1 we study the hit problem set up by f peterson of finding a minimal set of generators for pkpk as a module over the mod 2 steenrod algebra aa in this paper we study a minimal set of generators for aa module pkpk in some so called generic degrees and apply these results to explicitly determine the hit problem for k 4k 4

the cohomology of the steenrod algebra and representations of the general linear groups let tr k be the algebraic transfer that maps from the coinvariants of certain gl k representations to the cohomology of the steenrod algebra this transfer was defined by w singer as an algebraic version of the geometrical transfer tr k s bv k s s 0 it has been shown that the algebraic transfer is highly nontrivial more precisely that tr k is an isomorphism for k 1 2 3 and that tr k tr k is a homomorphism of algebras in this paper we first recognize the phenomenon that if we start from any degree d and apply sq repeatedly at most k 2 times then we get into the region in which all the iterated squaring operations are isomorphisms on the coinvariants of the gl k representations as a consequence every finite sq 0 family in the coinvariants has at most k 2 nonzero elements two applications are exploited the first main theorem is that tr k is not a n isomorphism for k 5 furthermore for every k 5 there are infinitely many degrees in which tr k is not an isomorphism we also show that if tr detects a nonzero element in certain degrees of ker sq 0 then it is not a monomorphism and further for each k tr k is not a monomorphism in infinitely many degrees the second main theorem is that the elements of any sq 0 family in the cohomology of the steenrod algebra except at most its first k 2 elements are either all detected or all not detected by tr k for every k applications of this study to the cases k 4 and 5 show that tr 4 does not detect the three families g d 3 and p and that tr 5 does not detect the family h n 1g n n 1

a proof of the conner floyd conjecture conjecture in is the annihilator ideal of tn the conjecture was proved forp odd by ravenel and wilson in 4 in this paper we give a much shorter proof valid for all primes briefly the proof runs as follows first of all we may replace mu by bp and vi by vi where bp z p v1 v2 and vo p moreover it is easy to see that in annihilates tn hence it is enough to show that the resulting map bp in bp tn is injective in section 1 we describe a comparatively small spectrum m n and a map f bp b z p n bp enm n such that f tn i 0 in section 2 we use the adams spectral sequence to show that bp m n is vn vn 1 9 torsion free which completes the proof aside from the adams spectral sequence the only result of any depth required for the proof is nakaoka s calculation 3 of the cohomology of symmetric products