Algebraic Geometric Invariants of Parafree Groups

Given a finitely generated (fg) group G, the set R(G) of homomorphisms from G to SL(2,C) inherits the structure of an algebraic variety known as the "representation variety" of G. This algebraic variety is an invariant of fg presentations of G. Call a group G parafree of rank n if it shares the lower central sequence with a free group of rank n, and if it is residually nilpotent. The deviation of a fg parafree group is the difference between the minimum possible number of generators of G and the rank of G. So parafree groups of deviation zero are actually just free groups. Parafree groups that are not free share a host of properties with free groups. In this paper algebraic geometric invariants involving the number of maximal irreducible components (mirc) of R(G) and the dimension of R(G) for certain classes of one-relator parafree groups are computed. It is then shown that in an infinite number of cases these invariants successfully discriminate between isomorphism types within the class of parafree groups of the same rank. This is quite surprising, since in this paper it also shown that an n generated group G is free of rank n iff Dim(R(G))=3n. In fact, a direct consequence of Theorem 1.6 in this paper is that given an arbitrary positive integer k, and any integer r > 1 there exist infinitely many non-isomorphic (fg) parafree groups of rank r and deviation one with representation varieties of dimension 3r, having more than k mirc of dimension 3r.Comment: 13 page

Varieties whose finitely generated members are free

We prove that a variety of algebras whose finitely generated members are free must be definitionally equivalent to the variety of sets, the variety of pointed sets, a variety of vector spaces over a division ring, or a variety of affine vector spaces over a division ring.Comment: 17 page

On two problems from "Hyperidentities and Clones"

A hyperidentity $E$ can be viewed as a statement in second order logic. When combined with a similarity type $\tau$, it can also be considered as a set of first order statements. Based on examples from "A small basis for hyperassociativity", which included hyperassociativity and $\tau=$, it was conjectured that each first order theory so produced was finitely axiomatizable. Part of the analysis suggested further investigating the relatively free 2-generated semigroup satisfying one or both of the equations $xxyxxyz=xxyyz$ and $zyyxx=zyxxyxx$. At ICM 1994, the conjecture above was refuted, and a finite basis problem arose: Is it decidable which pairs $$ give rise to finitely axiomatizable theories? This problem will be examined, and its connections to other fields (e.g. symbolic dynamics) will be reviewed. In doing so, we give partial solutions to problems 27 and 28 from "Hyperidentities and Clones"Comment: V2 will contain the official version. V1 is buggy. To be presented at Poster Session of ICM 2014 in Seou

On logically-geometric types of algebras

The connection between classical model theoretical types (MT-types) and logically-geometrical types (LG-types) introduced by B. Plotkin is considered. It is proved that MT-types of two $n$-tuples in two universal algebras coincide if and only if their LG-types coincide. An algebra $H$ is called logically perfect if for every two $n$-tuples in $H$ whose types coincide, one can be sent to another by means of an automorphism of this algebra. Some sufficient condition for logically perfectness of free finitely generated algebras is given which helps to prove that finitely generated free Abelian groups, finitely generated free nilpotent groups and finitely generated free semigroups are logically perfect. It is proved that if two Abelian groups have the same type and one of them is finitely generated and free then these groups are isomorphic.Comment: 12 page

Counting open negatively curved manifolds up to tangential homotopy equivalence

Under mild assumptions on a group G, we prove that the class of complete Riemannian n-manifolds of uniformly bounded negative sectional curvatures and with the fundamental groups isomorphic to G breaks into finitely many tangential homotopy types. It follows that many aspherical manifolds do not admit complete negatively curved metrics with prescribed curvature bounds.Comment: 22 pages, no figures; to appear in Journal of Differential Geometr

Quantizations of regular functions on nilpotent orbits

We study the quantizations of the algebras of regular functions on nilpotent orbits. We show that such a quantization always exists and is unique if the orbit is birationally rigid. Further we show that, for special birationally rigid orbits, the quantization has integral central character in all cases but four (one orbit in E_7 and three orbits in E_8). We use this to complete the computation of Goldie ranks for primitive ideals with integral central character for all special nilpotent orbits but one (in E_8). Our main ingredient is results on the geometry of normalizations of the closures of nilpotent orbits by Fu and Namikawa.Comment: 17 page

Strong solvability and residual finiteness for finitely decidable varieties

If V is a finitely generated variety such that the first-order theory of the finite members of V is decidable, we show that V is residually finite, and in fact has a finite bound on the sizes of subdirectly irreducible algebras. This result generalizes known results which assumed that V has modular congruence lattices. Our proof of the theorem in its full generality proceeds by showing that strongly solvable radicals of algebras in V are strongly abelian

Anabelian geometry with etale homotopy types

Anabelian geometry with etale homotopy types generalizes in a natural way classical anabelian geometry with etale fundamental groups. We show that, both in the classical and the generalized sense, any point of a smooth variety over a field k which is finitely generated over Q has a fundamental system of (affine) anabelian Zariski-neighbourhoods. This was predicted by Grothendieck in his letter to Faltings.Comment: 33 pages, refereed versio

Non-finitely based varieties of right alternative metabelian algebras

Since 1976, it is known from the paper by V. P. Belkin that the variety $\mathrm{RA_2}$ of right alternative metabelian (solvable of index 2) algebras over an arbitrary field is not Spechtian (contains non-finitely based subvarieties). In 2005, S. V. Pchelintsev proved that the variety generated by the Grassmann $\mathrm{RA_2}$-algebra of finite rank $r$ over a field $\mathcal{F}$, for $\mathrm{char}(\mathcal{F})\neq2$, is Spechtian iff $r=1$. We construct a non-finitely based variety $\mathfrak{M}$ generated by the Grassmann $\mathcal{V}$-algebra of rank $2$ of certain finitely based subvariety $\mathcal V\subset\mathrm{RA}_2$ over a field $\mathcal{F}$, for $\mathrm{char(\mathcal{F})}\neq2,3$, such that $\mathfrak{M}$ can also be generated by the Grassmann envelope of a five-dimensional superalgebra with one-dimensional even part.Comment: 20 page

Cosimplicial Groups and Spaces of Homomorphisms

Let $G$ be a real linear algebraic group and $L$ a finitely generated cosimplicial group. We prove that the space of homomorphisms $Hom(L_n,G)$ has a homotopy stable decomposition for each $n\geq 1$. When $G$ is a compact Lie group, we show that the decomposition is $G$-equivariant with respect to the induced action of conjugation by elements of $G$. The spaces $Hom(L_n,G)$ assemble into a simplicial space $Hom(L,G)$. When $G=U$ we show that its geometric realization $B(L,U)$, has a non-unital $E_\infty$-ring space structure whenever $Hom(L_0,U(m))$ is path connected for all $m\geq1$.Comment: 23 page