#### Contact Information

SC 1522(32)2-6657

m.sapir

#### Education

Ph.D., Moscow Pedagogical Institute, 1983

Employed at Vanderbilt Since: 1997

# Mark Sapir

Centennial Professor

#### Research Interests

Algorithmic problems in algebra, computers in education.

#### Publications

This page contains info about some of my publications. Most of the work done after 1992 has been supported by the National Science Foundation.

In May, 2004, we finished a large paper with Cornelia Drutu where we study asymptotic cones of groups and tree-graded spaces.

Several recent papers have been posted to the arXive. This includes our paper with Alexander Borisov where we prove that every ascendig HNN extension of a linear group is residually finite and some related results about periodic points of polynomial maps over finite fields.

In January 2003, we finished (with Victor Guba) a new paper about diagram groups. We proved that diagram groups can be realized as fundamental groups of positive paths of directed 2-complexs, computed homology groups and Poincare series of diagram groups over complete finite complexes, found F_infty diagram groups containing all countable diagram groups, etc. The paper is 42 pages long.

In October, 2002, we finished (with A.Yu. Olshanskii) a large paper where we proved that a finitely generated group G has solvable conjugacy problem if and only if it can be Frattini embedded into a finitely presented group with solvable conjugacy problem. This solves a problem by Collins (the paper is 120 pages long).

In December of 2000, A.Yu. Olshanskii and I finished a large paper where we construct the first finitely presented non-amenable group without free non-abelian subgroups. This group is torsion of exponent n>>1 by cyclic. So it satisfies the law [x,y]^n=1. This is a finitely presented counterexample to von Neumann's conjecture. What is possibly more important, this can be considered the first finitely presented "monster". Previously known "monsters" like a finitely generated infinite group with all subgroups cyclic of prime order and others (constructed by A.Yu. Olshanskii in 1979), are limits of hyperbolic groups and cannot be finitely presented. The paper is 105 pages long. It is going to appear in #96 of Publications of IHES, 2002.

Here is a shorter (10 pages) version of the previous paper. It is published Electronic Research Announcements (2002). No proofs but the main ideas are explained.

Here is our paper with Stuart Margolis and Pascal Weil about closed subgroups in pro-V-topologies of free groups. We prove that it is decidable whether a f.g. subgroup of a free group is closed in pro-nil topology. Thus we find an algorithm to decide given a set of partial permutations of a finite set whether this set is extendable to a nilpotent group of permutations of a finite set. It is still unknown if one can replace "nilpotent" by "solvable" in this statement. The length - 49 pages (published in IJAC 2001).

Here is our paper with Victor Guba where we prove that a diagram group contains a copy of the Thompson group F if and only if the corresponding directed complex contains a copy of the Dunce hat. Thus there exists a rigid connection between the Thompson group and the Dunce hat. The paper is 11 pages long (published in IJAC, 2002).

Here is our new paper with Victor Guba about Diagram groups where we prove that nilpotent subgroups of diagram groups are abelian, abelian subgroups are always free, the class of diagram groups is closed under the so called diagram product (free and direct products are particular cases of the diagram product), show that not every subgroup of a diagram group is a diagram group itself, find a distorted subgroup of the R. Thmopson group F, etc. The paper is 56 pages long. It is published in Mat. Sbornik (August 1999).

Here is a paper (joint with Olshanskii) where we present easy quasi-isometric embeddings of relatively free groups in finitely based varieties (in particular, the free Burnside groups) and Baumslag-Solitar groups into finitely presented groups with small Dehn functions. The paper is 23 pages long. It is published in Contemporary Mathematics, 264 (2000).

Here is my paper about amalgams of finite semigroups. I prove that the embeddability into any semigroup (ring) and the embeddability into finite semigroups (rings) of finite amalgams of semigroups (rings) is undecidable. The paper is published in J. Algebra (1999).

Here is the survey about Dehn functions and length distortion in groups that we have just finished writing with A.Yu. Olshanskii (Oct. 12, 1998, revised in August, 1999). It is 35 pages long. To see the HTML version of the survey, click here. The paper is published in IJAC (2000).

Paper "Quadratic isoperimetric functions for Heisenberg groups. A combinatorial proof" (joint with Olshanskii) is available here. The paper is only 8 pages long. It is published in Algebra, 11. J. Math. Sci. (New York) 93 (1999).

A paper (joint with Birget, Olshanskii and Rips) about isoperimetric functions of groups and computational complexity (finished in May 1998) is available here. In particular, we proved that a finitely generated group G has word problem in NP if and only if G is a subgroup of a finitely presented group H with polynomial Dehn function (moreover this subgroup has bounded length distortion). The paper is 47 pages long, it is poublished in Annals of Mathematics (2002).

This paper is based on the first paper (joint with Birget and Rips), finished in May 1997, where we described Dehn functions of groups. One of the results: if alpha is a number > 4 such that the first m digits of alpha can be

computed by a Turing machine in time < 2^{(2^m}) then n^{alpha} is equivalent to the Dehn function of a finitely presented group. Conversely, if n^{alpha} is the Dehn function of a finitely presented group then for every m the first m digits of alpha are computable in time < 2^{2^(2^m)}. The paper is 107 pages long. To download it, click here. The paper is published in the Annals of Mathematics (2002)

With Victor Guba, we wrote a paper about subnegative Dehn functions where we prove that the Dehn function of every non-trivial free product of groups is subnegative, that is satisfies the condition f(m+n)>= f(m)+f(n). My conjecture that every Dehn function of a finitely presented group is equivalent to a subnegative function is still open. The paper is published in Proc. AMS, 1999.

During the week of May 20-25 of 1996, we wrote two papers together with Stanislav Kublanovsky. In the paper "A variety with undecidable set of subalgebras of finite simple algebras" we construct a finitely based variety of algebras with two binary operations where the embeddability into finite simple algebras is undecidable (in semigroups, rings and groups this problem is always decidable). In the paper "Potential divisibility in finite semigroups is undecidable", we prove that there is no algorithm to decide, given a finite semigroup (associative ring) S and two elements a and b in S, whether there exists a bigger semigroup (associative ring) T>S where a divides b. This solves a 30 years old problem by John Rhodes. Both papers are published in IJAC, 1999.

Here is the list of some of my other papers. In order to download a paper, click on its name:

- Survey "Algorithmic problems in varieties" with Olga Kharlampovich (250 pp), IJAC ##4,5 1995.
- "Diagram groups" with Victor Guba (118 pp.), Memoirs of AMS, 1997.
- "Dehn functions and regular sets of normal forms for R. Thompson's group F", with Victor Guba, to appear in J. Austr. Math. Soc.
- "Quasi-identities of finite semigroups, symbolic dynamics, Burnside type problems", with Stuart Margolis, Israel J. Math. 92 (1995), 317-331
- Equations in finite semigroups and rings, J. Algebra, 1996.
- Some combinatorial results about indempotents in finite semigroups, with T. Hall, Descrete Math. Appl., 1996.
- The finite basis property for joins of semigroup varieties, with M. Volkov, published in Proc. AMS, 1993.
- Notes of my course "Combinatorics on words", preprint published by LITP, Paris (the book will be published by Birkhauser, so I removed the link)
- A survey on algorithmic problems (60+ pages), with Margolis and Meakin, Proc. NATO conference on Semigroups, Groups, Languages, 1995.
- Some undecidable and decidable problems for finite semigroups, with Kublanovsky, Hall, Margolis, Trotter, J. Pure and Applied Algebra, 1997.
- Irreducibility of semigroup pseudovarieties, with S.Margolis and P.Weil (published in IJAC, 2001)
- My CV including the list of all 100+ of my papers and books.

#### Awards

**American Mathematical Society Fellows** (2012)

**ICM Speakers** (2006)

**Lady Davis Fellowship** (2003)

**MSRI Awards** MSRI Simons Research Professorship (Fall 2007)

#### Students

- Curt Kent defended his PhD in March, 2013. He is working on topological properties of asymptotic cones of groups. Thesis: Topological properties of asymptotic cones of nitely generated groups.
- Iva Špakulová (Kozáková) defended her PhD in December, 2008. She is working on percolation in Cayley graphs of groups, and other probabilistic aspects of group theory. Thesis: "Percolation and Ising model on tree-like graphs".
- I was also a PhD advisor (together with John Meakin) of Tatyana Jajcayova. Thesis: "HNN extensions of inverse semigroups".

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