arXiv subjects
Publications and source records attributed to Grigori Avramidi.
For a group acting on a hyperbolic space, we set up an algorithm in the group algebra showing that ideals generated by few elements are free, where few is a function of the minimal displacement of the action, and derive algebraic, geometric, and topological consequences. In particular, we obtain lower bounds on Morse complexity of closed hyperbolic manifolds in terms of injectivity radius.
We find lower bounds on the topological complexity of fibers of PL and generic smooth maps $p:M^d\rightarrow\mathbb R^m$, where $M^d$ is a closed hyperbolic manifold of large injectivity radius. More precisely, we show that if the injectivity radius of $M$ is greater than $50\log((n+1)!)$, then for each dimension $0<k<d-m$ there is a point $z\in\mathbb R^m$ such that any cell structure on the fiber $p^{-1}(z)$ has more than $n$ cells of dimension $k$. The proof is based on a freedom theorem for ideals in group rings of hyperbolic groups proved in arXiv:2309.16791.
We systematically study L^2-Betti numbers in zero and prime characteristic and apply them to a conjecture of Wise stating that all towers of a finite 2-complex are non-positive if and only if the second L^2-Betti number vanishes.
We show that Gromov-Thurston branched covers satisfy the Singer conjecture whenever the degree of the cover is not divisible by a finite set of primes determined by the base manifold and the branch locus.
If $L$ is a flag triangulation of $S^{n-1}$, then the Davis complex $Σ_L$ for the associated right-angled Coxeter group $W_L$ is a contractible $n$-manifold. A special case of a conjecture of Singer predicts that the $L^2$-homology of such $Σ_L$ vanishes outside the middle dimension. We give conditions which guarantee this vanishing is preserved under edge subdivision of $L$. In particular, we verify Singer's conjecture when $L$ is the barycentric subdivision of the boundary of an $n$-simplex, and for general barycentric subdivisions of triangulations of $S^{2n-1}$. Using this, we construct explicit counterexamples to a torsion growth analogue of Singer's conjecture.
We introduce a hyperbolic reflection group trick which builds closed aspherical manifolds out of compact ones and preserves hyperbolicity, residual finiteness, and -- for almost all primes $p$ -- $\mathbb{F}_p$-homology growth above the middle dimension. We use this trick, embedding theory and manifold topology to construct Gromov hyperbolic $7$-manifolds that do not virtually fiber over a circle out of graph products of large finite groups.
We compute the mod $p$ homology growth of residual sequences of finite index normal subgroups of right-angled Artin groups. We find examples where this differs from the rational homology growth, which implies the homology of subgroups in the sequence has lots of torsion. More precisely, the homology torsion grows exponentially in the index of the subgroup. For odd primes $p$, we construct closed locally CAT(0) manifolds with nonzero mod $p$ homology growth outside the middle dimension. These examples show that Singer's conjecture on rational homology growth and Lück's conjecture on torsion homology growth are incompatible with each other, so at least one of them must be wrong.
We give examples of finite, simplicial $2$-complexes that do not PL embed in $\mathbb{R}^4$ and exhibit, for each such complex, a family of PL immersions into $\mathbb{R}^4$ that hide the obstruction to embedding in "higher and higher order Milnor invariants". In addition, we show that the embedding obstructions defined by Krushkal in [8] vanish for our examples. We also answer a question in a paper of Avramidi-Okun-Schreve.
We prove a division algorithm for group rings of high genus surface groups and use it to show that some $2$-complexes with surface fundamental groups are standard. We also give an application of division to cohomological dimension of $2$-relator groups acting on $\mathbb H^n$.
This paper accomplishes two things. First, we construct a geometric analog of the rational Tits building for general noncompact, complete, finite volume $n$-manifolds $M$ of bounded nonpositive curvature. Second, we prove that this analog has dimension less than $\lfloor n/2\rfloor$.
We give a simple construction of new, complete, finite volume manifolds $M$ of bounded, nonpositive curvature. These manifolds have ends that look like a mixture of locally symmetric ends of different ranks and their fundamental groups are not duality groups.
We find boundaries of Borel-Serre compactifications of locally symmetric spaces, for which any filling is incompressible. We prove this result by showing that these boundaries have small singular models and using these models to obstruct compressions. We also show that small singular models of boundaries obstruct $S^1$-actions (and more generally homotopically trivial $\mathbb Z/p$-actions) on interiors of aspherical fillings. We use this to bound the symmetry of complete Riemannian metrics on such interiors in terms of the fundamental group. We also use small singular models to simplify the proofs of some already known theorems about moduli spaces (the minimal orbifold theorem and a topological analogue of Royden's theorem).
In this paper we show that flat (m-1)-dimensional tori give nontrivial rational homology cycles in congruence covers of the locally symmetric space SL(m,Z) \SL(m,R)/SO(m). We also show that the dimension of the subspace of H_{m-1}(Γ\SL(m,R)/SO(m);Q) spanned by flat (m-1)-tori grows as one goes up in congruence covers.
We show that a finite type duality group of dimension $d>2$ is the fundamental group of a $(d+3)$-manifold with rationally acyclic universal cover. We use this to find closed manifolds with rationally acyclic universal cover and some nonvanishing $L^2$-Betti numbers outside the middle dimension, which contradicts a rational analogue of a conjecture of Singer.
We study complete, finite volume $n$-manifolds $M$ of bounded nonpositive sectional curvature. A classical theorem of Gromov says that if such $M$ has negative curvature then it is homeomorphic to the interior of a compact manifold-with-boundary, and we denote this boundary $\partial M$. If $n\geq 3$, we prove that the universal cover of the boundary $\widetilde{\partial M}$ and also the $π_1M$-cover of the boundary $\partial\widetilde M$ have vanishing $(n-2)$-dimensional homology. For $n=4$ the first of these recovers a result of Nguyen Phan saying that each component of the boundary $\partial M$ is aspherical. For any $n\geq 3$, the second of these implies the vanishing of the first group cohomology group with group ring coefficients $H^1(Bπ_1M;\mathbb Zπ_1M)=0$. A consequence is that $π_1M$ is freely indecomposable. These results extend to manifolds $M$ of bounded nonpositive curvature if we assume that $M$ is homeomorphic to the interior of a compact manifold with boundary. Our approach is a form of "homological collapse" for ends of finite volume manifolds of bounded nonpositive curvature. This paper is very much influenced by earlier, yet still unpublished work of Nguyen Phan.
The action dimension of a discrete group $Γ$ is the smallest dimension of a contractible manifold which admits a proper action of $Γ$. Associated to any flag complex $L$ there is a right-angled Artin group, $A_L$. We compute the action dimension of $A_L$ for many $L$. Our calculations come close to confirming the conjecture that if an $\ell^2$-Betti number of $A_L$ in degree $l$ is nonzero, then the action dimension of $A_L$ is $\ge 2l$.
Suppose X/Gamma is an arithmetic locally symmetric space of noncompact type (with the natural metric induced by the Killing form of the isometry group of X), and let p be a point on the visual boundary of X. It was shown by T.Hattori that if each horoball based at p intersects every Gamma-orbit in X, then p is not on the boundary of any Q-split flat in X (where Q is the field of rational numbers). We prove the converse. (This was conjectured by W.H.Rehn in some special cases.) Furthermore, we prove an analogous result when Gamma is a nonarithmetic lattice.
We study noncompact, complete, finite volume, Riemannian 4-manifolds $M$ with sectional curvature $-1<K<0$. We prove that $π_1 M$ cannot be a 3-manifold group. A classical theorem of Gromov says that $M$ is homeomorphic to the interior of a compact manifold $\M$ with boundary $\partial\barM$. We show that for each $π_1$-injective boundary component $C$ of $\M$, the map $i_*$ induced by inclusion $i\colon C\rightarrow \M$ has infinite index image $i_*(π_1 C)$ in $π_1 \M$. We also prove that $M$ cannot be homotoped to be contained in $\partial\M$.