Ignat Soroko

Department of Mathematics
University of North Texas
Denton, TX 76203-5017

Office: 410 GAB
Email: ignat.soroko@unt.edu

I am a Visiting Assistant Professor at the Department of Mathematics of the University of North Texas. I received my Ph.D. from the University of Oklahoma in May 2018 under the supervision of Noel Brady. After that I was a postdoc at Louisiana State University under the mentorship of Pallavi Dani, and a Dean's Postdoctoral Scholar at Florida State University.

Research interests:

Geometric group theory and low-dimensional topology, with emphasis on Artin groups, Coxeter groups, mapping class groups, quasi-isometry invariants, and Dehn functions of groups. I also have an ongoing interest in `Hanna Neumann' type of questions, stable commutator length, and criteria for linearity.

CV

Recent teaching evaluations: F22, S22, F21, S21 (online teaching), F19
Provost Certificate of Distinction in Teaching (OU, 2014)

Research

Published papers and preprints:

Divergence, thickness and hypergraph index for general Coxeter groups. (with P. Dani, Y. Naqvi and A. Thomas, submitted)
[ pdf ▪ arxiv ▪ GAP code ▪ slides ▪ video ▪ abstract ]

We introduce a computable combinatorial invariant, hypergraph index, for arbitrary Coxeter systems, which generalizes the construction of Levcovitz for right-angled Coxeter groups. We use it to obtain an upper bound on the order of divergence of general Coxeter groups. This upper bound is sharp for some infinite families of non-right-angled Coxeter groups, and conjecturally, for all Coxeter groups.

Property R_∞ for some spherical and affine Artin-Tits groups. (with M. Calvez) Journal of Group Theory 25, 6 (2022), 1045-1054
[ pdf ▪ arXiv ▪ journal ▪ zbMATH ▪ slides ▪ abstract ]

We give a short uniform proof of property R_∞ for the Artin-Tits groups of spherical types A_n, B_n, D₄, I₂(m), their pure subgroups, and for the Artin-Tits groups of affine types Ã_n-1 and C̃_n for n≥2.

Homological Dehn functions of groups of type FP₂. (with N. Brady and R. Kropholler, submitted)
[ pdf ▪ arXiv ▪ video ▪ abstract ]

We study the properties of homological Dehn functions of groups of type FP₂. 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 FP₂ with quartic homological Dehn function and unsolvable word problem.

Artin groups of types F₄ and H₄ are not commensurable with that of type D₄. Topology and its Applications 300 (2021), 107770
[ pdf ▪ arXiv ▪ journal ▪ MathReviews ▪ slides ▪ video ▪ abstract ]

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(D₄) 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
[ pdf ▪ arXiv ▪ journal ▪ MathReviews ▪ zbMATH ▪ abstract ]

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
30 (2020), no. 3, 625-666 [ pdf ▪ arXiv ▪ journal ▪ MathReviews ▪ zbMATH ▪ slides ▪ video ▪ abstract ]

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
Mathematics 142, 6 (2020), 1931-1944 [ pdf ▪ arXiv ▪ journal ▪ MathReviews ▪ slides ▪ abstract ]

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
[ pdf ▪ arXiv ▪ journal ▪ MathReviews ▪ zbMATH ▪ slides ▪ comments ▪ abstract ]

We generalize Culler's proof for the lower bound for the stable commutator length in free groups to the case of right-angled Artin groups.

Dehn functions of subgroups of right-angled Artin groups. (with N. Brady) Geometriae Dedicata 200 (2019), 197-239
[ pdf ▪ arXiv (old version) ▪ journal ▪ MathReviews ▪ comments ▪ abstract ]

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.

Other texts:

Linearity criteria for poly-free groups [ pdf ▪ abstract ]

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).

On a question of Peter Sarnak [ pdf ▪ abstract ]

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

Spring 2023: Calculus II, Section 310
Fall 2022: Linear Algebra and Vector Geometry, MATH 2700, Section 8

At FSU:

Spring 2022: Calculus with Analytic Geometry II, MAC 2312, Sections 1,7
Fall 2021: Calculus with Analytic Geometry II, MAC 2312, Sections 2,7

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 Saturday, April 1, 2023.