Volume comparison via boundary distances

The main subject of this expository paper is a connection between Gromov's filling volumes and a boundary rigidity problem of determining a Riemannian metric in a compact domain by its boundary distance function. A fruitful approach is to represent Riemannian metrics by minimal surfaces in a Banach space and to prove rigidity by studying the equality case in a filling volume inequality. I discuss recent results obtained with this approach and related problems in Finsler geometry.Comment: ICM 2010 sectional talk pape

Singularity Propagation for the Gurtin-Pipkin equation

We show that the Dirac delta function in the boundary condition of the Gurtin-Pipkin equation generates a moving delta-function with an exponentially decreasing factor.Comment: 7 pages, 13 reference

The intersection of subgroups in free groups and linear programming

We study the intersection of finitely generated subgroups of free groups by utilizing the method of linear programming. We prove that if $H_1$ is a finitely generated subgroup of a free group $F$, then the WN-coefficient $\sigma(H_1)$ of $H_1$ is rational and can be computed in deterministic exponential time in the size of $H_1$. This coefficient $\sigma(H_1)$ is the minimal nonnegative real number such that, for every finitely generated subgroup $H_2$ of $F$, it is true that $\bar {\rm r}(H_1, H_2) \le \sigma(H_1) \bar {\rm r}(H_1) \bar {\rm r}(H_2)$, where $\bar{ {\rm r}} (H) := \max ( {\rm r} (H)-1,0)$ is the reduced rank of $H$, ${\rm r} (H)$ is the rank of $H$, and $\bar {\rm r}(H_1, H_2)$ is the reduced rank of the generalized intersection of $H_1$ and $H_2$. We also show the existence of a subgroup $H_2^* = H_2^*(H_1)$ of $F$ such that $\bar {\rm r}(H_1, H_2^*) = \sigma(H_1) \bar {\rm r}(H_1) \bar {\rm r}(H_2^*)$, the Stallings graph $\Gamma(H_2^*)$ of $H_2^*$ has at most doubly exponential size in the size of $H_1$ and $\Gamma(H_2^*)$ can be constructed in exponential time in the size of $H_1$.Comment: 27 pages, 2 figures. arXiv admin note: text overlap with arXiv:1607.0305

The bounded and precise word problems for presentations of groups

We introduce and study the bounded word problem and the precise word problem for groups given by means of generators and defining relations. For example, for every finitely presented group, the bounded word problem is in NP, i.e., it can be solved in nondeterministic polynomial time, and the precise word problem is in PSPACE. The main technical result of the paper states that, for certain finite presentations of groups, which include the Baumslag-Solitar one-relator groups and free products of cyclic groups, the bounded word problem and the precise word problem can be solved in polylogarithmic space. As consequences of developed techniques that can be described as calculus of brackets, we obtain polylogarithmic space bounds for the computational complexity of the diagram problem for free groups, for the width problem for elements of free groups, and for computation of the area defined by polygonal singular closed curves in the plane. We also obtain polynomial time bounds for these problems.Comment: 94 pages, 33 figure

On Bousfield's problem for solvable groups of finite Pr\"ufer rank

For a group $G$ and $R=\mathbb Z,\mathbb Z/p,\mathbb Q$ we denote by $\hat G_R$ the $R$-completion of $G.$ We study the map $H_n(G,K)\to H_n(\hat G_R,K),$ where $(R,K)=(\mathbb Z,\mathbb Z/p),(\mathbb Z/p,\mathbb Z/p),(\mathbb Q,\mathbb Q).$ We prove that $H_2(G,K)\to H_2(\hat G_R,K)$ is an epimorphism for a finitely generated solvable group $G$ of finite Pr\"ufer rank. In particular, Bousfield's $HK$-localisation of such groups coincides with the $K$-completion for $K=\mathbb Z/p,\mathbb Q.$ Moreover, we prove that $H_n(G,K)\to H_n(\hat G_R,K)$ is an epimorphism for any $n$ if $G$ is a finitely presented group of the form $G=M\rtimes C,$ where $C$ is the infinite cyclic group and $M$ is a $C$-module

Intersecting free subgroups in free products of left ordered groups

A conjecture of Dicks and the author on rank of the intersection of factor-free subgroups in free products of groups is proved for the case of left ordered groups.Comment: 11 page

On the Burnside problem on periodic groups

It is proved that the free $m$-generated Burnside groups $\Bbb{B}(m,n)$ of exponent $n$ are infinite provided that $m>1$, $n\ge2^{48}$.Comment: 4 page

On a conjecture of Imrich and M\"uller

A conjecture of Imrich and M\"uller on rank of the intersection of subgroups of free groups is disproved.Comment: 4 pages, 1 figur

On joins and intersections of subgroups in free groups

We study graphs of (generalized) joins and intersections of finitely generated subgroups of a free group. We show how to disprove a lemma of Imrich and M\"uller on these graphs and how to repair this lemma.Comment: 14 pages, 4 figure

Linear programming and the intersection of free subgroups in free products of groups

We study the intersection of finitely generated factor-free subgroups of free products of groups by utilizing the method of linear programming. For example, we prove that if $H_1$ is a finitely generated factor-free noncyclic subgroup of the free product $G_1 * G_2$ of two finite groups $G_1$, $G_2$, then the WN-coefficient $\sigma(H_1)$ of $H_1$ is rational and can be computed in exponential time in the size of $H_1$. This coefficient $\sigma(H_1)$ is the minimal positive real number such that, for every finitely generated factor-free subgroup $H_2$ of $G_1 * G_2$, it is true that $\bar {\rm r} (H_1, H_2) \le \sigma(H_1) \bar {\rm r}(H_1) \bar {\rm r}(H_2)$, where $\bar{ {\rm r}} (H) = \max ( {\rm r} (H)-1,0)$ is the reduced rank of $H$, ${\rm r}(H)$ is the rank of $H$, and $\bar {\rm r}(H_1, H_2)$ is the reduced rank of the generalized intersection of $H_1$ and $H_2$. In the case of the free product $G_1 * G_2$ of two finite groups $G_1$, $G_2$, it is also proved that there exists a factor-free subgroup $H_2^* = H_2^*(H_1)$ such that $\bar {\rm r}(H_1, H_2^*) = \sigma(H_1) \bar {\rm r}(H_1) \bar {\rm r}(H_2^*)$, $H_2^*$ has at most doubly exponential size in the size of $H_1$, and $H_2^*$ can be constructed in exponential time in the size of $H_1$.Comment: 53 pages, 2 figure