Department of Mathematics
University of North Texas
Denton, TX 76203-5017
Office: 410 GAB
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.
We give a short uniform proof of property R∞ for the Artin-Tits groups of spherical types An, Bn, D4, I2(m), their pure subgroups, and for the Artin-Tits groups of affine types Ãn-1 and C̃n for n≥2.
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.
[ pdf ▪ arXiv ▪ doi ▪ 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.
[ pdf ▪ arXiv ▪ doi ▪ 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).
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 ▪ doi ▪ 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 ▪ doi ▪ 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) ▪ doi ▪ 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.
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.
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