arXiv subjects

Boris Lishak

Publications and source records attributed to Boris Lishak.

6 recordsLinked to original sources

arXiv2021

Filling metric spaces

We prove a new version of isoperimetric inequality: Given a positive real $m$, a Banach space $B$, a closed subset $Y$ of metric space $X$ and a continuous map $f:Y \rightarrow B$ with $f(Y)$ compact $$\inf_FHC_{m+1}(F(X))\leq c(m)HC_m(f(Y))^{\frac{m+1}{m}},$$ where $HC_m$ denotes the $m$-dimensional Hausdorff content, the infimum is taken over the set of all continuous maps $F:X\longrightarrow B$ such that $F(y)=f(y)$ for all $y\in Y$, and $c(m)$ depends only on $m$. Moreover, one can find $F$ with a nearly minimal $HC_{m+1}$ such that its image lies in the $C(m)HC_m(f(Y))^{1\over m}$-neighbourhood of $f(Y)$ with the exception of a subset with zero $(m+1)$-dimensional Hausdorff measure. The paper also contains a very general coarea inequality for Hausdorff content and its modifications. As an application we demonstrate an inequality conjectured by Larry Guth that relates the $m$-dimensional Hausdorff content of a compact metric space with its $(m-1)$-dimensional Urysohn width. We show that this result implies new systolic inequalities that both strengthen the classical Gromov's systolic inequality for essential Riemannian manifolds and extend this inequality to a wider class of non-simply connected manifolds.

math.DG↗

arXiv2019

Complexity of Unknotting of Trivial 2-knots

We construct families of trivial $2$-knots $K_i$ in $\mathbb{R}^4$ such that the maximal complexity of $2$-knots in any isotopy connecting $K_i$ with the standard unknot grows faster than a tower of exponentials of any fixed height of the complexity of $K_i$. Here we can either construct $K_i$ as smooth embeddings and measure their complexity as the ropelength (a.k.a the crumpledness) or construct PL-knots $K_i$, consider isotopies through PL knots, and measure the complexity of a PL-knot as the minimal number of flat $2$-simplices in its triangulation. These results contrast with the situation of classical knots in $\mathbb{R}^3$, where every unknot can be untied through knots of complexity that is only polynomially higher than the complexity of the initial knot.

math.MG↗

arXiv2017

Sizes of spaces of triangulations of 4-manifolds and balanced presentations of the trivial group

Let $M$ be any compact four-dimensional PL-manifold with or without boundary (e.g. the four-dimensional sphere or ball). Consider the space $T(M)$ of all simplicial isomorphism classes of triangulations of $M$ endowed with the metric defined as the minimal number of bistellar transformations required to transform one of two considered triangulations into the other. Our main result is the existence of an absolute constant $C>1$ such that for every $m$ and all sufficiently large $N$ there exist more than $C^N$ triangulations of $M$ with at most $N$ simplices such that pairwise distances between them are greater than $2^{2^{\ldots^{2^N}}}$ ($m$ times). This result follows from a similar result for the space of all balanced presentations of the trivial group. ("Balanced" means that the number of generators equals to the number of relations). This space is endowed with the metric defined as the minimal number of Tietze transformations between finite presentations. We prove a similar exponential lower bound for the number of balanced presentations of length $\leq N$ with four generators that are pairwise $2^{2^{\ldots^{2^N}}}$-far from each other. If one does not fix the number of generators, then we establish a super-exponential lower bound $N^{const\ N}$ for the number of balanced presentations of length $\leq N$ that are $2^{2^{\ldots^{2^N}}}$-far from each other.

math.GT↗

arXiv2015

Automorphisms of the Baumslag-Gersten group

We classify homomorphisms of the Baumslag-Gersten group into itself. We prove it is Hopfian and co-Hopfian. We show that the group of outer automorphisms of the Baumslag-Gersten group is isomorphic to the dyadic rationals with the addition operation. These results are not new. They were obtained by Andrew M. Brunner in \cite{Brunner}. However, our exposition is self-contained and, hopefully, more accessible for some readers.

math.GR↗

arXiv2015

Balanced presentations of the trivial group and four-dimensional geometry

We prove that 1) There exist infinitely many non-trivial codimension one "thick" knots in $\mathbb{R}^5$; 2) For each closed four-dimensional smooth manifold $M$ and for each sufficiently small positive $ε$ the set of isometry classes of Riemannian metrics with volume equal to $1$ and injectivity radius greater than $ε$ is disconnected; 3) For each closed four-dimensional $PL$-manifold $M$ and any $m$ there exist arbitrarily large values of $N$ such that some two triangulations of $M$ with $<N$ simplices cannot be connected by any sequence of $<M_m(N)$ bistellar transformations, where $M_m(N)=\exp(\exp(\ldots \exp (N)))$ ($m$ times).

math.MG↗

arXiv2015

Balanced finite presentations of the trivial group

We construct a sequence of balanced finite presentations of the trivial group with two generators and two relators with the following property: The minimal number of relations required to demonstrate that a generator represents the trivial element grows faster than the tower of exponentials of any fixed height of the length of the finite presentation.

math.GR↗