# Mark Sapir

**Mark Sapir** (born February 12, 1957)[1] is a U.S. and Russian mathematician working in geometric group theory, semigroup theory and combinatorial algebra. He is a Centennial Professor of Mathematics in the Department of Mathematics at Vanderbilt University.

Mark V. Sapir | |
---|---|

Born | February 12, 1957 |

Nationality | U.S. |

Alma mater | Ural State University |

Known for | research in geometric group theory |

Scientific career | |

Fields | Mathematics |

Institutions | Vanderbilt University |

Doctoral advisor | Lev Shevrin |

## Biographical and professional information

Sapir received his undergraduate degree in mathematics (diploma of higher education) from the Ural State University in Yekaterinburg (then called Sverdlovsk), Russia, in 1978.[1] He received his PhD in mathematics (Candidate of Sciences) degree, joint from the Ural State University and Moscow State Pedagogical Institute in 1983, with Lev Shevrin as the advisor.[1]

Afterwards Sapir has held faculty appointments at the Ural State University, Sverdlovsk Pedagogical Institute, University of Nebraska at Lincoln, before coming as a professor of mathematics to Vanderbilt University in 1997. He was appointed a Centennial Professor of Mathematics at Vanderbilt in 2001.

Sapir gave an invited talk at the International Congress of Mathematicians in Madrid in 2006.[2] He gave an AMS Invited Address at the American Mathematical Society Sectional Meeting in Huntsville, Alabama in October 2008.[3] He gave a plenary talk at the December 2008 Winter Meeting of the Canadian Mathematical Society.[4] Sapir gave the 33d William J. Spencer Lecture at the Kansas State University in November 2008.[5] He gave the 75th KAM Mathematical Colloquium lecture at the Charles University in Prague in June 2010.[6]

Sapir became a member of the inaugural class of Fellows of the American Mathematical Society in 2012.[7]

Sapir founded the *Journal of Combinatorial Algebra*, published by the European Mathematical Society, and has served as its founding editor-in-chief since 2016.[8] He is also currently an editorial board member for the journals Groups, Complexity, Cryptology and Algebra and Discrete Mathematics. His past editorial board positions include Journal of Pure and Applied Algebra, Groups, Geometry, and Dynamics, Algebra Universalis, and International Journal of Algebra and Computation (as Managing Editor).

A special mathematical conference in honor of Sapir's 60th birthday took place at the University of Illinois at Urbana–Champaign in May 2017.[9]

Mark Sapir's elder daughter, Jenya Sapir, is also a mathematician, currently an assistant professor in the Department of Mathematics of Binghamton University.[10]

Mark Sapir and his wife Olga Sapir became naturalized U.S. citizens in July 2003,[11] after suing the BCIS in federal court over a multi-year delay of their citizenship application originally filed in 1999.[12]

## Mathematical contributions

Sapir's early mathematical work concerns mostly semigroup theory.

In geometric group theory his most well-known and significant results are obtained in two papers published in the Annals of Mathematics in 2002,[13][14] the first joint with Jean-Camille Birget and Eliyahu Rips, and the second joint with Birget, Rips and Aleksandr Olshansky. The first paper provided an essentially complete description of all the possible growth types of Dehn functions of finitely presented groups. The second paper proves that a finitely presented group has the word problem solvable in non-deterministic polynomial time (**NP**) if and only if this group embeds as a subgroup of a finitely presented group with polynomial Dehn function. A combined featured review of these two papers in Mathematical Reviews characterized them as ``remarkable foundational results regarding isoperimetric functions of finitely presented groups and their connections with the complexity of the word problem".[15]

Sapir is also known for his work, mostly joint with Cornelia Drutu, on developing the asymptotic cone approach to the study of relatively hyperbolic groups.[16][17]

A 2002 paper of Sapir and Olshansky constructed the first known finitely presented counter-examples to the Von Neumann conjecture.[18]

Sapir also introduced, in a 1993 paper with Meakin,[19] the notion of a diagram group, based on finite semigroup presentations. He further developed this notion in subsequent joint papers with Guba.[20] Diagram groups provided a new approach to the study of Thompson groups, which appear as important examples of diagram groups.

## Selected publications

- Guba, Victor; Sapir, Mark (1997). "Diagram groups".
*Memoirs of the American Mathematical Society*.**130**(620). doi:10.1090/memo/0620. MR 1396957. - Sapir, Mark V.; Birget, Jean-Camille; Rips, Eliyahu (2002). "Isoperimetric and isodiametric functions of groups".
*Annals of Mathematics*. Second Series.**156**(2): 345–466. arXiv:math/9811106. doi:10.2307/3597196. MR 1933723. - Birget, Jean-Camille; Ol'shanskii, Alexander Yu.; Rips, Eliyahu; Sapir, Mark V. (2002). "Isoperimetric functions of groups and computational complexity of the word problem".
*Annals of Mathematics*. Second Series.**156**(2): 467–518. arXiv:math/9811105. doi:10.2307/3597195. MR 1933724. - Olʹshanskii, Alexander Yu.; Sapir, Mark V. (2002). "Non-amenable finitely presented torsion-by-cyclic groups".
*Publications Mathématiques de l'IHÉS*.**96**(2003): 43–169. MR 1985031. - Borisov, Alexander; Sapir, Mark (2005). "Polynomial maps over finite fields and residual finiteness of mapping tori of group endomorphisms".
*Inventiones Mathematicae*.**160**(2): 341–356. arXiv:math/0309121. doi:10.1007/s00222-004-0411-2. MR 2138070. - Drutu, Cornelia; Sapir, Mark (2008). "Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups".
*Advances in Mathematics*.**217**(3): 1313–1367. doi:10.1016/j.aim.2007.08.012. MR 2383901.

## References

- Mark Sapir's CV, Department of Mathematics, Vanderbilt University. Accessed November 4, 2018
- ICM Plenary and Invited Speakers, International Mathematical Union. Accessed November 4, 2018.
- AMS Sectional Meeting Invited Addresses. 2008 Fall Southeastern Meeting Huntsville, AL, October 24-26, 2008 (Friday – Sunday) Meeting #1044. American Mathematical Society. Accessed November 4, 2018.
- Plenary lectures, December 2008 Winter Meeting, Canadian Mathematical Society. Accessed November 4, 2018.
- William J. Spencer Lectures, Department of Mathematics, Kansas State University. Accessed November 4, 2018.
- KAM Mathematical Colloquia, Department of Applied Mathematics, Charles University. Accessed November 4, 2018.
- List of Fellows of the American Mathematical Society, American Mathematical Society. Accessed November 4, 2018.
- Editorial Board,
*Journal of Combinatorial Algebra*. European Mathematical Society. Accessed November 4, 2018. - CONFERENCE ON GEOMETRIC AND COMBINATORIAL METHODS IN GROUP THEORY. In honor of Mark Sapir's 60th birthday Department of Mathematics, University of Illinois at Urbana–Champaign. Accessed November 4, 2018.
- Jenya Sapir's webpage, Department of Mathematics of Binghamton University. Accessed November 4, 2018.
- Sapir vs Aschcroft, Case No. 3:03-0326 (Middle District of Tenn. 2003), Judge Aleta A. Trauger. Order of August 13, 2003. LexisNexis. Accessed November 11, 2018.
- Jim Patterson, Russian couple file lawsuit over INS delay.
*Plainview Daily Herald*, April 24, 2003. Accessed November 11, 2018. - Birget, J.-C.; Ol'shanskii, A. Yu; Rips, E.; Sapir, M. V. (September 2002). "Isoperimetric Functions of Groups and Computational Complexity of the Word Problem".
*Annals of Mathematics*. Second Series.**156**(2): 467. arXiv:math/9811106. doi:10.2307/3597196. MR 1933723. - Sapir, Mark V.; Birget, Jean-Camille; Rips, Eliyahu (September 2002). "Isoperimetric and Isodiametric Functions of Groups".
*Annals of Mathematics*. Second Series.**156**(2): 345. arXiv:math/9811105. doi:10.2307/3597195. MR 1933724. - Ilya Kapovich (2005)
*Mathematical Reviews*, MR1933723 and MR1933724. - Druţu, Cornelia; Sapir, Mark (September 2005). "Tree-graded spaces and asymptotic cones of groups".
*Topology*.**44**(5): 959–1058. doi:10.1016/j.top.2005.03.003. MR 2153979. - Druţu, Cornelia; Sapir, Mark V. (February 2008). "Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups".
*Advances in Mathematics*.**217**(3): 1313–1367. doi:10.1016/j.aim.2007.08.012. MR 2383901. - Olʹshanskii, Alexander Yu.; Sapir, Mark V. (2002). "Non-amenable finitely presented torsion-by-cyclic groups".
*Publications Mathématiques de l'IHÉS*(96): 43–169. MR 1985031. - Meakin, John; Sapir, Mark (1993). "Congruences on free monoids and submonoids of polycyclic monoids".
*Journal of the Australian Mathematical Society, Series A*.**54**(2): 236–253. MR 1200795. - Guba, Victor; Sapir, Mark (1997). "Diagram groups".
*Memoirs of the American Mathematical Society*.**130**(620). doi:10.1090/memo/0620. MR 1396957.