Integrated Glofit and Ibrutinib as Novel Immune Therapy Evaluation in Treatment-Naive Mantle Cell Lymphoma (IGNITE MCL): A Phase Ib/II Study of Glofitamab Plus Ibrutinib With Obinutuzumab Pretreatment in MCL Patients ≥ 65 or Ages 18-64 With High-Risk Features

* Ability to understand the purpose and risks of the study and to provide signed informed consent * Pathologically confirmed MCL, with documentation of chromosome translocation t(11;14)(q13;q32) and/or overexpression of cyclin D1 in association with other relevant markers (e.g., CD5, CD19, CD20, PAX5). Cyclin D1 negative or t(11,14) negative/SOX 11 positive MCL patients can be enrolled if eligibility criteria are otherwise satisfied * Age 18-64 with one or more of the following poor risk features defined as: * High risk mutational variants including p53 aberrations (mutation\[s\] by next generation sequencing \[NGS\] and/or 17p deletion), KMT2D, NSD2, NOTCH1,CDKN2A, NOTCH2, SMARCA4, and CCND1; * Blastoid or pleomorphic phenotype; * Complex karyotype with ≥ 3 abnormalities (in addition to t(11,14)) on routine karyotyping; * Ki67 \> 30%; * High risk Mantle Cell Lymphoma International Prognostic Index (MIPI) score ≥ 6.2; and/or * p53 expression on immunohistochemistry (IHC), defined as ≥ 50% * Age ≥ 65. For this population, no poor risk features are required to be eligible * No prior systemic anticancer therapies for MCL * Presence of radiologically measurable lymphadenopathy and/or extranodal lymphoid malignancy * Able to provide biosamples for MRD testing and pathology. If fresh tissue is not available, archival samples can be used for some assessments at the discretion of the investigator * Eastern Cooperative Oncology Group (ECOG) performance status of ≤ 2 * Absolute neutrophil count (ANC) ≥ 1.0 × 10\^9/L independent of growth factor support * Platelets ≥ 100 × 10\^9/L (≥ 50 × 10\^9/L if bone marrow \[BM\] involvement), independent of transfusion support in either situation * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) ≤ 3 × upper limit of normal (ULN) * Total bilirubin ≤ 1.5 × ULN (unless due to Gilbert's syndrome or of non hepatic origin) * Estimated creatinine clearance ≥ 50 mL/min by Cockcroft Gault method * Undetectable hepatitis B virus (HBV) surface antigen (sAg) serology. Confirmed by polymerase chain reaction (PCR) for HBV deoxyribonucleic acid (DNA) if results are disputable * Undetectable hepatitis C virus (HCV) antigen serology. Confirmed by PCR for HCV ribonucleic acid (RNA) if results are disputable * Undetectable HIV based on serology. Enrollment will be considered if HIV is controlled with treatment (i.e., undetectable viral load for 6 months prior and the CD4 counts is ≥ 200/µL). Such patients must be willing to modify HIV therapy while on-treatment and during applicable wash-out periods, as needed, to address drug-drug interactions * Willing and able to participate in all required evaluations and procedures in this protocol including swallowing capsules without difficulty * Persons of childbearing potential (PCBP): Persons of childbearing potential must have a negative serum (beta-human chorionic gonadotropin \[b-hCG\]) or urine pregnancy test at Screening and be willing to use approved contraception while on treatment and for the longest following, applicable time period: 18 months after the last dose of obinutuzumab, 2 months after the last dose of glofitamab, 3 months for tocilizumab, or 1 month after the last dose of ibrutinib. Women who are pregnant or breastfeeding are ineligible for this study * Persons that produce viable sperm: Willingness to use approved contraception while on-treatment and for the longest applicable time period following: 6 months after the last dose of obinutuzumab or 2 months after the last dose of glofitamab, tocilizumab, or ibrutinib * Willingness to not breastfeed or donate ova or sperm: If obinutuzumab was the last study drug received, the participant must wait for 6 months. Otherwise, patients must agree to wait 3 months for sperm and 1 month for ova donations (and to breastfeed) after the last dose other study drugs * The effects of GLIB on the developing human fetus are unknown. Should a participant or participant's sexual partner become pregnant or suspect a pregnancy while participating in this study, the individual should inform their treating physician immediately Exclusion Criteria: * Previous MCL-directed treatment. Treatment with corticosteroids (up to 20 mg dexamethasone or equivalent daily) is allowed prior to and during the screening period for patients with aggressive clinical behavior. All steroids used for disease control must be discontinued within 7 days before starting study treatment except for doses ≤ 20 mg per day of prednisone or equivalent. Ongoing steroids as premedications or for cytokine release syndrome (CRS) management are allowed on study * Clinically significant cardiovascular disease such as uncontrolled or symptomatic arrhythmias, congestive heart failure, or myocardial infarction within 6 months of screening, or any class 3 (moderate) or class 4 (severe) cardiac disease as defined by the New York Heart Association Functional Classification * History of prior malignancy except for the following: * Malignancy treated with curative intent and with no evidence of active disease present for more than 2 years before screening and felt to be at low risk for recurrence by treating physician * Persons with low grade prostate cancer on a watch and wait strategy are eligible * Adequately treated lentigo maligna melanoma without current evidence of disease or adequately controlled nonmelanomatous skin cancer * Adequately treated carcinoma in situ without current evidence of disease * Ongoing hormonal therapy alone for prior malignancy is allowed * Concurrent use of cytotoxic chemotherapy; radiotherapy; immunotherapy; hormone therapy (other than contraceptives, hormone-replacement therapy, or megestrol acetate); and biologic agents (other than hematopoietic growth factors, if clinically indicated and used in accordance with manufacture and Investigator recommendations), unless approved by the investigator * Received systemic immunosuppressive medications (e.g., cyclophosphamide, azathioprine, methotrexate, thalidomide, and anti-tumor necrosis factor agents) within 2 weeks prior to obinutuzumab infusion with the exception of those described * Known history of hypersensitivity to * Humanized or murine monoclonal antibodies or products * A CD3 and / or CD20 antibody * Glofitamab * Ibrutinib * Tocilizumab * Current or past history of epilepsy, central nervous system (CNS) vasculitis, and neurodegenerative disease * History of autoimmune disease (e.g., myocarditis, pneumonitis, myasthenia gravis, myositis, autoimmune hepatitis, systemic lupus erythematosus, rheumatoid arthritis, inflammatory bowel disease, vascular thrombosis associated with antiphospholipid syndrome, Wegener's granulomatosis, Sjögren's syndrome, Guillain-Barre syndrome, multiple sclerosis, vasculitis, or glomerulonephritis. Patients with a remote history of, or well controlled, autoimmune disease may be eligible to enroll after consultation with the primary investigator * History of bleeding risks: * Stroke, transient ischaemic attack (TIA),or intracranial hemorrhage within 2 years of first dose of study drug given no remaining neurological deficits * Known bleeding diathesis (e.g., hemophilia or von Willebrand disease) * Presence of a gastrointestinal ulcer diagnosed by endoscopy within 3 months prior to cycle 1 day 1 (C1D1) * Requires or receiving anticoagulation with warfarin or equivalent vitamin K antagonists (e.g., phenprocoumon) within 7 days of first dose of C1D1 * Requires treatment with a strong CYP3A inhibitor/inducer, except for the following: * A plan to modify concurrent CYP3A inhibitor/inducer and/or wash-out periods prior to C1D1 * Topical ketoconazole: Based on its low overall bioavailability, there are no restrictions * Concurrent participation in another therapeutic clinical trial * History of confirmed progressive multifocal leukoencephalopathy (PML) or lymphomatous involvement of the CNS * Known or suspected history of hemophagocytic lymphohistiocystosis (HLH) * Evidence of ongoing acute or systemic infections (bacterial, fungal, or viral), or any major episode of infection requiring hospitalization or treatment with IV antibiotics (for IV antibiotics this pertains to completion of last course of antibiotic treatment) within 4 weeks of dosing, except localized fungal infections of skin or nails. Subjects may be receiving prophylactic antiviral or antibacterial therapies at the discretion of the investigator * Receipt of live vaccine within 4 weeks of enrollment or during study treatment period * Any life-threatening illness, medical condition, or organ system dysfunction which, in the investigator's opinion, could compromise the subject's safety, interfere with the absorption or metabolism of ibrutinib capsules, or put the study outcomes at undue risk