Geometric group theory and low-dimensional topology, with emphasis on Artin
groups, Coxeter groups, free-by-cyclic groups, quasi-isometric invariants, Dehn functions of groups, and stable commutator length. I also
have an ongoing interest in `Hanna Neumann' type of questions, mapping class groups, profinite groups, and criteria for linearity.
We study the properties of homological Dehn functions of groups of type FP_{2}. We show how to build uncountably many quasi-isometry classes of such groups with a given homological Dehn function. As an application we prove that there exists a group of type FP_{2} with quartic homological Dehn function and unsolvable word problem.
Artin groups of types F_{4} and H_{4} are not commensurable with that of type D_{4}.
Topology and its Applications 300 (2021), 107770,
We resolve two out of six cases left undecided in a recent article of Cumplido and Paris. We also determine the automorphism group of Art(D_{4}) and describe torsion elements, their orders and conjugacy classes in all Artin groups of spherical type modulo their centers.
Linearity of some low-complexity mapping class groups. Forum Mathematicum 32 (2020), no. 2, 279-286.
We show that the pure mapping class group of the orientable surface of genus g with b boundary components and n punctures is linear for the following values of (g,b,n): (0,m,n), (1,2,0), (1,1,1), (1,0,2), (1,3,0), (1,2,1), (1,1,2), (1,0,3).
A (longer) earlier version with an alternative computation "from first principles": pdf.
Realizable ranks of joins and intersections of subgroups in free groups.
International Journal of Algebra and Computation
We describe the locus of possible ranks ( rk(H∨K), rk(H∩K) ) for any given subgroups H, K of a free group. In particular, we resolve the remaining open case (m=4) of R.Guzman's "Group-Theoretic Conjecture" in the affirmative.
Uncountably many quasi-isometry classes of groups of type FP.
(with R. Kropholler and
I. Leary)
American Journal of
We prove that among I. Leary's groups of type FP there exist uncountably many non-quasi-isometric ones. We also prove that for each n≥4 there exist uncountably many quasi-isometry classes of non-finitely presented n-dimensional Poincare duality groups.
Genus bounds in right-angled Artin groups.
(with M. Forester and J. Tao)
Publicacions Matemàtiques 64 (2020), no. 1, 233-253.
We show that polynomials of arbitrary integer degree are realizable as Dehn functions of subgroups in right-angled Artin groups. In the Appendix we prove that no finite index subgroup of the famous Gersten's free-by-cyclic group can be embedded into a right-angled Artin group.
In his article on what is now called 'Lubotzky's linearity criterion', Alexander Lubotzky established a criterion for a group Aut(F) to be linear, where F is a free group of finite rank in terms of 'p-congruence systems'. We generalize this result of his to the case of groups of the form A(semidirect)F, where A is a finitely generated subgroup of Aut(F).
In his 2012 MSRI `Notes on thin groups' Peter Sarnak asks if a specific pair of symplectic matrices generates an infinite index subgroup in Sp(4,Z). We approach this question with a technique adapted from mapping class groups.
Teaching:
All materials related to teaching are on Canvas and WebAssign.
Fall 2021: Calculus with Analytic Geometry II, MAC 2312, Sections 2,7
Past teaching at LSU:
Spring 2021: Math 1552, sections 5,6 (Calculus-II, online)
Fall 2019: Math 1552, section 1 (Calculus-II)
Fall 2018: Math 1550, section 37 (Calculus-I)
Personal:
My wife Hayat Hokayem is an Associate Professor
in the College of Education at Texas Christian University, Fort Worth, TX.
Our son Vikenty (born in 2014): 1, 2, 3, 4,
5, 6, 7.
Our son Nikolai (born in 2017): 1, 2,
3, 4, 5,
6, 7.
Vikenty and Nikolai together: 1.
This page was last updated on Friday, December 17, 2021.